Self-Decomposability and the Lambert Law

Abstract

The Lambert W is a Bernstein function and hence the cumulant function of a positive infinitely divisible law, the Lambert law $L(\Lambda )$. The Lambert law is self-decomposable with a unit rate compound Poisson background driving Lévy process (BDLP) with a certain jump law $L\left(J\right)$. Thus, it satisfies the random perpetuity relation $\Lambda \stackrel{L}{=}U(\Lambda +J)$, where U has the standard uniform law, as well as the relation $\Lambda \stackrel{L}{=}U\mathcal{S}J$, where $\mathcal{S}$ is the stationary-excess operator. If the law of J is unspecified, then these two in-law relations together characterise the law of $\Lambda$. Generalisations are pursued in which the Poisson jump rate a is arbitrary. If $a\ne 1$, the above two conditions determine a family of positive infinitely divisible laws for which the Laplace–Stieltjes transforms satisfy an algebraic trinomial equation investigated by Lambert and Euler. These laws occur as limiting distributions of shot noise processes and continuous-time branching processes. A sequence of n-times self-decomposable positive laws derived from a given subordinator is called a BDLP ladder. The BDLP ladder that includes the Lambert law has cumulant functions which are polynomial forms of the Lambert function. Under moment conditions, general BDLP ladder laws are asymptotically normal or stable.

1. Introduction

We begin by recalling some facts. The cumulant function (CF) of the law $L\left(X\right)$ of a non-negative random variable is defined as $C\left(\theta \right)=-\log E\left(e^{-\theta X}\right)$, $\theta \ge 0$. The law $L\left(X\right)$ is infinitely divisible (infdiv) if $C\left(\theta \right)$ is a Bernstein function (written $C\in \mathcal{B}$), meaning that there is a constant $\delta \ge 0$ and a (Lévy) measure $\mathcal{L}\left(dx\right)$ on $(0,\infty )$ satisfying $\mathcal{L}\left(\right\{0\left\}\right)=0$ and ${\int }_0^{\infty }(x\wedge 1)\mathcal{L}\left(dx\right)<\infty$ such that

$$C\left(\theta \right)=\delta \theta +{\int }_0^{\infty }\left(1-e^{-\theta x}\right)\mathcal{L}\left(dx\right).$$

(1)

In other words, $C^{\prime }\left(\theta \right)$ is completely monotone.

If $X\stackrel{L}{=}cX+X_c$ for all $c\in (0,1]$, where on the right-hand side X and $X_c$ are independent, then $L\left(X\right)$ is called self-decomposable (SD). An analytic equivalence is that $C\in \mathcal{B}$ and $\lambda \left(dx\right)=\ell \left(x\right)dx$, with $x\ell \left(x\right)$ non-increasing in $(0,\infty )$. A particular case is where $x\ell \left(x\right)$ is completely monotone, with a representing measure $\nu \left(dx\right)$ (called the Thorin measure) satisfying ${\int }_{0+}^{1+}\log v\nu \left(v\right)>-\infty$ and ${\int }_{1+}^{\infty }{y}^{-1}\nu \left(dy\right)<\infty$. We say that $C\left(\theta \right)$ is Thorin–Bernstein, $C\in \mathcal{T}\mathcal{B}$, and the corresponding law $L\left(X\right)$ is a generalised gamma convolution (GGC). The GGCs comprise the smallest class containing gamma laws while being closed under convolution and convergence in law. In recent decades, the infdiv nature of many laws has been exposed by showing they are GGCs; see, e.g., pp. 378–387 in [1] and the many examples in [2].

In this paper, we are concerned with some structural issues involving a special GGC, namely, the Lambert law. The Lambert W function is defined as a solution of the equation

$$W{e}^W=\theta .$$

(2)

A real-valued concave increasing solution $W\left(\theta \right)$ exists which maps $[-e^{-1},\infty )$ onto $[-1,\infty )$. There is a second decreasing negative-valued solution $W_1\left(\theta \right)$ which maps $[-e^{-1},0)$ onto $[-1,-\infty )$. We are concerned with the first of these, the principal solution. These solutions are respectively denoted by $W_p$ and $W_m$ in §4.13 of the standard reference [3]; see Corless et al. [4] for the fundamental properties of the Lambert W function and a description of some of its applications, [5] for a summary of some applications, and [6] for an extensive classification and listing of applications. Many of these occurrences of W arise because solving equations describing the situation at hand produces a functional equation which can be manipulated into the Lambert form (2).

A more recent insight has revealed that W has an integral representation which links to probability theory, specifically $W\in \mathcal{B}$, i.e.,

$$W\left(\theta \right)={\int }_0^{\infty }\left(1-e^{-\theta x}\right)\omega \left(x\right)dx,$$

(3)

where $\omega$ is the density of a Lévy measure. Consequently, there is a corresponding infdiv law $L(\Lambda )$, the Lambert law, for which the Laplace–Stieltjes transform (LST) is

$$\lambda \left(\theta \right):=E\left(e^{-\theta \Lambda }\right)=e^{-W\left(\theta \right)}.$$

(4)

It follows from (2) that $\lambda \left(\theta \right)=W\left(\theta \right)/\theta$, and hence that $\Lambda$ has the probability density function (pdf)

$$\ell \left(x\right)={\int }_x^{\infty }\omega \left(y\right)dy,(x>0).$$

This is a pdf because

$${\int }_0^{\infty }\ell \left(x\right)dx={\int }_0^{\infty }y\omega \left(y\right)dy=W^{\prime }\left(0\right)=1.$$

Refining the assertion $W\in \mathcal{B}$, it is known that $W\in \mathcal{T}\mathcal{B}$, and hence that the Lambert law is SD. There are many contributions leading to this outcome, and we briefly review them in Section 2.

Our point of departure is the stochastic integral representation enjoyed by any SD random variable X. In summary, X can be expressed as a random Laplace–Stieltjes transform in which the integrator function is a Lévy process, which conventionally is dubbed the background driving Lévy process (BDLP); see [7] or [8] for details. This representation for the Lambert law is

$$\Lambda \stackrel{L}{=}{\int }_0^{\infty }{e}^{-t}d{B}_t,$$

(5)

where the BDLP $\left(B_t\right)$ is a compound Poisson with jumps that occur at unit rate and have the exponential mixture density

$${\omega }_B\left(y\right)={\int }_0^{\infty }v{e}^{-yv}\nu \left(dv\right).$$

(6)

This representation entails a characterisation result (Theorem 1) for $L(\Lambda )$ as the unique solution up to scaling satisfying the random perpetuity relation $X\stackrel{L}{=}U(X+J)$ and subject to $X\stackrel{L}{=}U\mathcal{S}J$, where U has a standard uniform law, J has the jump density (6), $\mathcal{S}$ is the stationary excess operator (defined below), and the random variables on the right-hand sides of these relations are independent. This convention applies throughout the paper; random variables occurring in the right-hand side of stochastic representation formulae are mutually independent. We also investigate the generalisation obtained by allowing the BDLP in (5) to have an arbitrary rate a while retaining the jump law (6) (Theorems 2 and 3).

J.H. Lambert (1728–1777) is credited with being the first to treat a functional equation. He solved $y-y^m=q$ for y as a power series in q. Later, Euler expressed Lambert’s equation in a more symmetric form and derived the Lambert functional equation as a limiting case; see [4]. We express Lambert’s version as $x^{-1/a}-x^{-1}=(1-a)\theta$ with $\theta \ge 0$ and $a>0$. This degenerates when $a=1$, but can be understood for all positive a by expressing it as $\tau \left(x\right)=\theta$, where for $0<s\le 1$ we have

$$\tau \left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{s^{-1/a}-s^{-1}}{1-a}}\hfill & \mathrm{if}a\ne 1,\hfill \\ -s^{-1}\log s\hfill & \mathrm{if}a=1.\hfill \end{array}\right.$$

(7)

The function satisfies $\tau (0+)=\infty$, $\tau \left(1\right)=0$ and $\tau$ is convex-decreasing. Hence, there is a unique convex-decreasing solution $\chi \left(\theta \right)$ on $[0,\infty )$ satisfying $\chi \left(0\right)=1$; additionally, $\chi (\infty -)=0$. If $a=1$, then $\chi \left(\theta \right)={\theta }^{-1}W\left(\theta \right)$, the LST of the Lambert law. One may wonder whether $\chi$ is the LST of a probability law. We show in Section 4 that the answer is yes; in fact it specifies an infdiv law, embedding the Lambert law in a one-parameter family.

Our starting point in Section 4 is the above pair of in-law equalities (i.e., (15) and (16) below) where the uniform factor U is replaced with $U^{1/a}$, where a is a positive constant. Equivalently, the BDLP in (5) has rate a and a more general jump law. With these alterations, the perpetuity relation has a scale family of infdiv solutions, so the question is whether there is a jump law which is compatible with the stationary excess constraint. We find that the two conditions lead to a first-order differential equation, the solution of which solves the above Lambert trinomial equation, (25) below. It turns out that this solution is indeed the LST of an infdiv law which has a nice stochastic representation. The principal existence results, comprising Theorem 4 and Remark 2, follow from Lemma 2. Corollary 1 exhibits a duality between the cases $a<1$ and $a>1$. Theorem 6 shows that the laws so constructed are in fact GGCs, implying that the Lambert law is not a singular instance. Corollary 2 provides yet another proof that $W\in \mathcal{T}\mathcal{B}$. Section 4 concludes with some tail estimates for $L\left(X_a\right)$, a relation with shot-noise processes and the BDLP representation, alternative random variable representations, and the specific cases $a=1/2$, its dual $a=2$, and $a=1/3$.

The BDLP representation of an SD law on the line can be iterated to give a sequence of SD laws $\{L\left(X_n\right):n\ge 1\}$, where for $n\ge 1$ the Lévy process determined by $L\left(X_{n-1}\right)$ is the BDLP of $L\left(X_n\right)$. This construction determines an operator $\mathcal{I}:L\left(X_{n-1}\right)\mapsto L\left(X_n\right)$ (Iksanov et al. [9] and references therein), and is equivalent to an earlier formulation of Urbanik [10] in terms of limit laws of normed sums of independent random variables. The continued iteration of $\mathcal{I}$ requires that $E\left[\right(\log |1+X_0{\left|\right)}^n]<\infty$ for all $n\ge 1$, called the log-condition. Starting with a given infdiv law $L\left(X_0\right)$, we refer to the sequence $\{{\mathcal{I}}^n\left(L\left(X_0\right)\right):n\ge 0\}$ as a BDLP ladder, reflecting the nesting relation $L_{n+1}\subset L_n$, where $L_0$ comprises all infdiv laws $L\left(X_0\right)$ satisfying the log-condition and $L_n={\mathcal{I}}^n\left(L_0\right)$. Elements of $L_n$ are termed n-times SD laws in [9] (p. 1364). The collective term ‘multiply SD’ is used in other papers where the concept has been studied in very general settings. Note too that the operator $\mathcal{I}$ is often denoted by $\Phi$.

We stress here that our indexing $L_0,L_1,\dots$ is a unit shift from that in the cited formulations. Here, laws in $L_n$ could be termed as order n-SD, i.e., SD laws are order 1-SD. In addition, expressions obtained for our examples and results have a tidier form.

Our interest lies in the subsets $L_n^{+}={\mathcal{I}}^n\left(L_0^{+}\right)$, where $L_0^{+}$ comprises positive infdiv laws which, with no loss of generality, have no drift components; see Berg and Forst [11] for some representation results. These authors worked with an effectively equivalent formulation in terms of sets ${\mathcal{B}}_n$ ($n\ge 0$) of CFs $C_n\left(\theta \right)$, where ${\mathcal{B}}_0=\{C\left(\theta \right)\in \mathcal{B}\}$ and $C_n\left(\theta \right)\in {\mathcal{B}}_n$ if, for each $\alpha \in (0,1)$, there is a CF $C_{\alpha }\left(\theta \right)\in {\mathcal{B}}_{n-1}$ such that $C_n\left(\theta \right)=C_n\left(\alpha \theta \right)+C_{\alpha }\left(\theta \right)$. We define BDLP ladders more closely in Section 5 and illustrate the concept with the gamma and Lambert ladders. In particular, the Lambert ladder, which can be defined by $X_1=\Lambda$, provides a sequence of polynomial forms in $W\left(\theta \right)$ which are Bernstein functions.

A little general theory for BDLP ladders is developed in Section 6 which supplements the representation results in [11]. If $a:=E\left(X_0\right)<\infty$, then $E\left(X_n\right)=a$ for all n and the Formula (50) for the Lévy density of $L\left(X_n\right)$ implies the length-bias representation (54) generalizing a representation for positive infdiv laws developed in [12]. Theorem 9 asserts a random variable interpretation for (50) which is used to show that $L\left(X_n\right)$ is asymptotically degenerate (Theorem 10) with normal fluctuations around the limit (Theorem 11).

It is known that the limit set $L_{\infty }^{+}={\cap }_{n\ge 0}{L}_n^{+}$ is the smallest set containing positive stable laws while being closed under convolution and weak convergence. The CFs of its members have the mixture form ${\int }_0^1{\theta }^{\alpha }d\sigma \left(\alpha \right)$, where $\sigma$ is a finite measure on $(0,1]$ (Theorem 1.6 in [11]). The positive stable laws correspond to point masses. In Theorem 12, we use a general stochastic representation of the CF $C_n\left(\theta \right)$ of $L\left(X_n\right)$ under a fairly strong regular variation condition on $C_0\left(\theta \right)$ to show that there is a norming sequence $b_n\to 0$ such that $\left\{b_n{X}_n\right\}$ converges to a positive stable law. Examples 7 and 8 alter this condition in ways which cause a partial failure of the proof of Theorem 12. However, convergence to a stable law still holds, though with a rather different form of norming sequence. There is room for more work on the limiting behaviour of $L\left(X_n\right)$. BDLP ladders are related in Lemma 5 and Theorems 13 and 14 to Urbanik’s [10] representations for spectrally positive laws in $L_n$.

We show in Section 7 how the laws developed in Section 4 arise as so-called martingale limit laws for supercritical continuous-time branching processes.

Some notation changes from section to section, but no confusion should arise.

2. $\mathit{W}\in \mathcal{B}$: A Review

Two proofs yielding $W\in \mathcal{B}$ are based on the derivative formula

$$W^{\left(n\right)}\left(\theta \right)={(-1)}^{n-1}{\displaystyle \frac{{\mathcal{P}}_n\left(W\left(\theta \right)\right)\exp (-nW\left(\theta \right))}{{(1+W\left(\theta \right))}^{2n-1}}},$$

(8)

where ${\mathcal{P}}_n\left(x\right)$ is a polynomial of degree $n-1$ which satisfies a certain recursion relation. A very simple argument [13] shows that these polynomials are positive-valued in $(e^{-1},\infty )$, and hence that $W^{\prime }\left(\theta \right)$ is completely monotone, whence (3). Kalugin and Jeffrey [14] independently established the same result by showing that the coefficients of each polynomial ${\mathcal{P}}_n$ form a positive sequence which is log-concave and unimodal.

Another pair of proofs allow us to deduce from Stieltjes transform representations that $W\in \mathcal{T}\mathcal{B}$. We first observe that (2) and (8) with $n=1$ yield

$$W^{\prime }\left(\theta \right)={\displaystyle \frac{W\left(\theta \right)}{\theta (1+W(\theta \left)\right)}}.$$

(9)

Pakes [13] used a complex variable argument to show that the left-hand side has the integral representation

$$W^{\prime }\left(\theta \right)={\int }_0^{\infty }{\displaystyle \frac{\nu \left(dv\right)}{\theta +v}},$$

(10)

giving the above representation, and also that

$${\theta }^{-1}W\left(\theta \right)={\int }_{e^{-1}}^{\infty }{\displaystyle \frac{N\left(y\right)}{y(\theta +y)}}dy,$$

(11)

where $N\left(x\right)=\nu [0,x]$ is a distribution function supported in $[e^{-1},\infty )$. Independent related arguments [15] yield the above second representation with the identification $N\left(y\right)={\pi }^{-1}\Im W(-y)$, and (10) follows as a consequence.

The coefficients in the power series expansion $W^{\prime }\left(\theta \right)={\sum }_{n\ge 1}{c}_n{(-\theta )}^{n-1}$, where $c_n=n^n/n!$, have an integral representation (Nuttall’s integral), which yields another demonstration [16] that $W\in \mathcal{T}\mathcal{B}$. This approach proceeds from an explicit algebraic expression for the inverse (i.e., quantile) function of $N\left(y\right)$ in (11) (see (1.12) in [16]), implying that N has a pdf $n\left(y\right)$ such that

$$W^{\prime }\left(\theta \right)={\int }_{e^{-1}}^{\infty }{\displaystyle \frac{n\left(y\right)}{\theta +y}}dy.$$

The left-hand side is the Laplace transform of $x\omega \left(x\right)$; hence,

$$x\omega \left(x\right)={\int }_{e^{-1}}^{\infty }{e}^{-xy}n\left(y\right)dy.$$

(12)

Asymptotic estimates for $N\left(y\right)$ and $n\left(y\right)$ when $y\gg 1$ and $y\approx e^{-1}$ are further outcomes.

Finally, it was shown in [17] how $W\in \mathcal{T}\mathcal{B}$ follows from results about continuous branching processes (CBPs). Implicit in that paper is possibly the shortest proof that $W\in \mathcal{B}$. This follows from the fact that any CBP $(Z_t:t\ge 0)$ is constructed as a certain random-time transformation of a spectrally positive Lévy process $\left(Y_t\right)$. Interpreting $Z_t$ as the mass of a randomly changing lump of material at time t, the characteristic branching property is that components of a finite partition of the lump evolve independently with the same overall process dynamics. This implies that $L\left(Z_t\right)$ is infdiv for each t. Assuming that $Z_0=1$ and that the log-growth rate $m=t^{-1}\log E\left(Z_t\right)\in (0,\infty )$, the process is supercritical. Then, there exists a function ${\eta }_t\downarrow 0$ as $t\uparrow \infty$ such that ${\eta }_t{Z}_t\stackrel{a.s.}{\to }M$ (the so-called martingale limit), which is almost surely finite non-degenerate and infdiv.

If $C\left(\theta \right)$ is the CF of $\left(Y_t\right)$, then the CF $\Omega \left(\theta \right)$ of M solves the differential equation ${\Omega }^{\prime }\left(\theta \right)=C(\Omega \left(\theta \right))$. An admissible form is $\Omega \left(\theta \right)=\theta /(1+\theta )$, i.e., $\left(Y_t\right)$ is the compound Poisson process with unit jump rate and jumps have the standard exponential law. In this case, the differential equation for $\Omega$ is the same as (11). Since $W\left(0\right)=\Omega \left(0\right)=0$, it follows that $\Omega \left(\theta \right)=W\left(\theta \right)$, whence $W\in \mathcal{B}$. This example satisfies some general conditions for CBPs implying that $L\left(M\right)$ is SD; see Theorem 4.1 in [17].

We end this section with some results about the infdiv innovation law $L\left(I_c\right)$ in the autoregression representation $\Lambda \stackrel{L}{=}c\Lambda +I_c$. Its CF is

$$W_c\left(\theta \right)={\int }_0^{\infty }\left(1-e^{-\theta x}\right){\omega }_c\left(x\right)dx=W\left(\theta \right)-W\left(c\theta \right),$$

where from (3) and (12) we have

$$\begin{array}{ccc}\hfill {\omega }_c\left(x\right)& :=& \omega \left(x\right)-c^{-1}\omega (x/c)=x^{-1}{\int }_0^{\infty }\left(e^{-xy}-c^{-1}{e}^{-xy/c}\right)n\left(y\right)dy\hfill \\ & =& x^{-1}{\int }_0^{\infty }{e}^{-xy}(n\left(y\right)-cn\left(cy\right))dy.\hfill \end{array}$$

(13)

Lemma 1. 

(a) The innovation law $L\left(I_c\right)$ is compound-Poisson; ${\int }_0^{\infty }{\omega }_c\left(dy\right)=W_c(\infty -)=A:=-\log c<\infty$.

(b) The Lévy density ${\omega }_c\left(x\right)$ satisfies the asymptotic relation as $x\downarrow 0$,

$${\int }_0^{x}y{\omega }_c\left(y\right)dy\sim {\displaystyle \frac{Ax}{{\left(\log x\right)}^2}}.$$

Proof. 

Assertion (a) follows from the large $\theta$ expansion, $W\left(\theta \right)=\log \theta -\left(\log \log \theta \right)(1+O(1/\log \theta \left)\right)$. Similarly, using (9), as $\theta \to \infty$ we have

$$\begin{array}{ccc}\hfill W_c^{\prime }\left(\theta \right)& =& {\int }_0^{\infty }x{\omega }_c\left(x\right)e^{-\theta x}dx=W^{\prime }\left(\theta \right)-c{W}^{\prime }\left(c\theta \right)\hfill \\ & =& {\theta }^{-1}{\displaystyle \frac{W\left(\theta \right)-W\left(c\theta \right)}{(1+W(\theta \left)\right)(1+W(c\theta \left)\right)}}\sim {\displaystyle \frac{A}{\theta {\left(\log \theta \right)}^2}}.\hfill \end{array}$$

Assertion (b) follows from the (Tauberian) Theorem 1.7.1′ in [18]. □

Formal differentiation in part (b) suggests that $x{\omega }_c\left(x\right)\sim A{\left(\log x\right)}^{-2}$. This is supported by a tedious calculation which yields

$$\theta W_c^{\prime }\left(\theta \right)={\displaystyle \frac{A}{W^2\left(\theta \right)}}+{\displaystyle \frac{A^2-2A}{W^3\left(\theta \right)}}\left(1+o\left(1\right)\right).$$

Observe that (13) exhibits $x{\omega }_c\left(x\right)$ as a Laplace transform. However, $n\left(y\right)-cn\left(cy\right)$ must change sign within its support, because if it is non-negative then ${\omega }_c\left(x\right)$ would be the Lévy density of a Thorin–Bernstein function and hence $L\left(I_c\right)$ could not be compound-Poisson.

3. The Random Perpetuity Representation

Jurek [8] explains how the BDLP representation (5) yields a random perpetuity representation. Let ${\tau }_n$ denote the n-th jump time of the compound Poisson BDLP. Since ${\tau }_n$ is a stopping time with respect to $\left(B_t\right)$, we have

$$\Lambda =e^{-{\tau }_n}\left(J_n+{\int }_{0+}^{\infty }{e}^{-t}d{B}_{t+{\tau }_n}\right).\stackrel{L}{=}{V}_n\left(J_n+{\tilde{\Lambda }}_n\right),$$

(14)

where ${\tilde{\Lambda }}_n$ is an independent copy of $\Lambda$, $V_n=e^{-{\tau }_n}$, $J_n=B\left({\tau }_n\right)$ and is equal to the sum of n independent copies of $J_1$, and $V_n$, $J_n$, and ${\tilde{\Lambda }}_n$ are all independent. In addition, $V_n$ has density $v_n\left(u\right)={(-\log u)}^{n-1}/n!$, where $0<u\le 1$. The random difference Equation (14) comprises the perpetuity relations satisfied by the Lambert law.

From now on, we consider only the case with $n=1$, writing (14) as

$$X\stackrel{L}{=}U(X+J),$$

(15)

where $U=V_1$ has the standard uniform law, $J=J_1$, and the random variables on the right-hand side are independent. Thus, $L(\Lambda )$ satisfies (15) and $E(\Lambda )=E\left(J\right)$. Since $E(\Lambda )=W^{\prime }\left(0\right)=1$, we have $E\left(J\right)=1$.

We need the following preliminaries for Theorem 1 below, which characterises the Lambert law: if X has a distribution function (DF) F supported in $[0,\infty )$ and a finite mean $\mu >0$, then we define the stationary excess DF corresponding to F as

$$\left(\mathcal{S}F\right)\left(x\right)={\mu }^{-1}{\int }_0^x\overline{F}\left(y\right)dy,$$

where $\mathcal{S}$ is called the stationary-excess operator; see Whitt [19]. In addition, $\mathcal{S}X$ denotes any random variable having the DF $\mathcal{S}F$. If $\psi \left(\theta \right)=E\left(e^{-\theta X}\right)$ denotes the LST of X, then the LST of $\mathcal{S}X$ is $(1-\psi (\theta \left)\right)/\left(\mu \theta \right)$. The LST of the product $UX$, where the factors are independent and U has the standard uniform law, is ${\theta }^{-1}{\int }_0^{\theta }\psi \left(y\right)dy$. Finally, $({\Lambda }_t:t\ge 0)$ denotes the Lambert subordinator; $E\left(e^{-\theta {\Lambda }_t}\right)=e^{-tW\left(\theta \right)}$. Recall the definition of ${\omega }_B$ in (6).

Theorem 1. 

(a) The Lambert law $L(\Lambda )$ satisfies the perpetuity relation (15) and also satisfies $\Lambda \stackrel{L}{=}U\mathcal{S}J$. In addition, $J\stackrel{L}{=}{\Lambda }_{ϵ}$, where ϵ has the standard exponential law and is independent of $\left({\Lambda }_t\right)$.

(b) Conversely, suppose that J is a non-negative random variable with a finite mean $\mu >0$. If X satisfies (15) and the stationary-excess condition

$$X\stackrel{L}{=}U\mathcal{S}J,$$

(16)

then $X\stackrel{L}{=}\mu \Lambda$ and $J/\mu$ has the density ${\omega }_B$.

Proof. 

If J has the density ${\omega }_B$, then its LST is

$$j\left(\theta \right)={\widehat{\omega }}_B\left(\theta \right)={\int }_0^{\infty }{\displaystyle \frac{z}{\theta +z}}\nu \left(dz\right).$$

Hence, the LST of $\mathcal{S}J$ is

$${\displaystyle \frac{1-j\left(\theta \right)}{\theta }}={\int }_0^{\infty }{\displaystyle \frac{\nu \left(dz\right)}{\theta +z}}=W^{\prime }\left(\theta \right),$$

(17)

using (10). Thus, the LST of $U\mathcal{S}J$ is ${\theta }^{-1}{\int }_0^{\theta }{W}^{\prime }\left(y\right)dy=W\left(\theta \right)/\theta$, i.e., $\Lambda \stackrel{L}{=}U\mathcal{S}J$. This proves the direct assertion. Note that in light of (9) and (17) the above LST of J can be expressed as

$$j\left(\theta \right)={\left(1+W\left(\theta \right)\right)}^{-1},$$

(18)

implying the second assertion of (a).

To prove the converse, let $x\left(\theta \right)$ and $j\left(\theta \right)$ denote the LSTs of X and J in the perpetuity relation (15); note that its LST version is

$$x\left(\theta \right)={\theta }^{-1}{\int }_0^{\theta }x\left(y\right)j\left(y\right)dy.$$

Writing $w\left(\theta \right)=\theta x\left(\theta \right)$, we have $w^{\prime }\left(\theta \right)=w\left(\theta \right)j\left(\theta \right)/\theta$. The condition in (16) yields the relation

$$w\left(\theta \right)={\int }_0^{\theta }{\displaystyle \frac{1-j\left(y\right)}{\mu y}}dy.$$

Differentiating and equating the two forms of $w^{\prime }$ yields

$$j\left(\theta \right)={\left(1+\mu w\left(\theta \right)\right)}^{-1},$$

(19)

and hence the differential equation

$$w^{\prime }\left(\theta \right)={\displaystyle \frac{w\left(\theta \right)}{\theta (1+\mu w(\theta \left)\right)}}.$$

Separating variables and integrating yields $\log w\left(\theta \right)+\mu w\left(\theta \right)=\log \left(K\theta \right)$, where K is an arbitrary constant. Rearranging gives $w\left(\theta \right)\exp \left(\mu w\right(\theta \left)\right)=K\theta$, so it follows from (2) that $x\left(\theta \right)=W\left(K\mu \theta \right)/\left(\mu \theta \right)$. However, $W^{\prime }\left(0\right)=1$ together with the context requires that $x\left(0\right)=1$; thus, it follows that $K=1$, and hence that $X=\mu \Lambda$, as asserted. The assertion about $L\left(J\right)$ follows by comparing (18) and (19). □

As motivation for the next section, we may ask which laws we obtain by running the compound Poisson process in (5) at rate $a>0$? The answer is given by the following result, which extends Theorem 1.

Theorem 2. 

(a) If $a>0$ and

$${\Lambda }_a:={\int }_0^{\infty }{e}^{-t}d{B}_{at}={\int }_0^{\infty }{e}^{-t/a}d{B}_t,$$

then $L\left({\Lambda }_a\right)$ is the one-dimensional law of the Lambert subordinator at time a. It satisfies the perpetuity relation

$${\Lambda }_a\stackrel{L}{=}{U}_a({\Lambda }_a+J),$$

(20)

where $U_a:=U^{1/a}$ has the density function $a{u}^{a-1}$ for $0<u<1$ and $L\left(J\right)$ has the LST (18).

(b) If $a\ge 1$, then the extended stationary excess relation holds:

$${\Lambda }_a\stackrel{L}{=}{U}_a({\Lambda }_{a-1}+\mathcal{S}J),(a\ge 1).$$

(21)

Proof. 

Referring to (17), the CF of $\left(B_{at}\right)$ is $a(1-j\left(\theta \right))=a\theta W^{\prime }\left(\theta \right)$, and hence the CF of the SD law $L\left({\Lambda }_a\right)$ is $aW\left(\theta \right)$, proving the first assertion. The perpetuity relation (20) follows in the same way as (14).

Next, since ${\Lambda }_0\stackrel{L}{=}0$, (21) indeed reduces to (16) if $a=1$. To prove the general relation, observe first that if R is a positive random variable with LST $r\left(\theta \right)$ and independent of $U_a$, then the LST of the product $U_{a}R$ is

$$E\left[r\left(U_a\theta \right)\right]=a{\int }_0^{1}r\left(u\theta \right)u^{a-1}=a{\theta }^{-a}{\int }_0^{\theta }r\left(y\right)y^{a-1}dy.$$

Assuming that R can be chosen so that ${\Lambda }_a\stackrel{L}{=}{U}_{a}R$, we can equate this product LST with ${(W\left(\theta \right)/\theta )}^a$, the LST of ${\Lambda }_a$, to obtain $W^a\left(\theta \right)=a{\int }_0^{\theta }r\left(y\right)y^{a-1}dy$. Differentiation yields $r\left(\theta \right)={(W\left(\theta \right)/\theta )}^{a-1}{W}^{\prime }\left(\theta \right)$. The first factor is the LST of ${\Lambda }_{a-1}$ if $a\ge 1$, so in view of (17) we see that $R\stackrel{L}{=}{\Lambda }_{a-1}+\mathcal{S}J$, and (21) follows. □

If a is sufficiently large, then the relation (21) can be iterated to express ${\Lambda }_a$ in terms of ${\Lambda }_{⟨a⟩}$, where the subscript denotes the fractional part of a. For a trivial example, if $a=2$, then ${\Lambda }_{⟨a⟩}=0$ and

$${\Lambda }_2\stackrel{L}{=}{U}_{1,2}(\mathcal{S}{J}_1+U_{2,2}\mathcal{S}{J}_2),$$

where for $i=1,2$ we have $U_{i,2}$ as independent copies of $\sqrt{U}$ and $J_i$ as independent copies of J.

Canceling the factors $U_a$ between (20) and (21) yields the balance identity

$${\Lambda }_a+J\stackrel{L}{=}{\Lambda }_{a-1}+\mathcal{S}J.$$

The designation is explained by the fact that ${\Lambda }_a$ is stochastically increasing in a; equating the identities (15) (with $X=\Lambda$) and (16) shows that $\mathcal{S}J\stackrel{L}{=}J+\Lambda$, i.e., $\mathcal{S}J$ is stochastically larger than J.

We end this section with the following generalisation of (20). Given $p\in [0,1]$, let $I_p$ denote a random variable having the Bernoulli $\left(p\right)$ law, i.e., $P(I_p=1)=p$.

Theorem 3. 

Let γ be a positive constant, $a\in (0,\gamma ]$ and $p=a/\gamma$. Then,

$${\Lambda }_a\stackrel{L}{=}{U}_{\gamma }({\Lambda }_a+I_{p}J).$$

Proof. 

Following the idea leading to (21), we seek a random variable R such that ${\Lambda }_a\stackrel{L}{=}{U}_{\gamma }R$. As above, this produces the identity ${\theta }^{\gamma -a}{W}^a\left(\theta \right)=\gamma {\int }_0^{\theta }r\left(y\right)y^{\gamma -1}dy$, and hence

$$r\left(\theta \right)=(1-p){\left({\displaystyle \frac{W\left(\theta \right)}{\theta }}\right)}^a+p{W}^{\prime }\left(\theta \right){\left({\displaystyle \frac{W\left(\theta \right)}{\theta }}\right)}^{a-1}={\left({\displaystyle \frac{W\left(\theta \right)}{\theta }}\right)}^a\left[1-p+p{\displaystyle \frac{\theta W^{\prime }\left(\theta \right)}{W\left(\theta \right)}}\right].$$

It follows from (9) that

$${\displaystyle \frac{\theta W^{\prime }\left(\theta \right)}{W\left(\theta \right)}}={\displaystyle \frac{1}{1+W\left(\theta \right)}}=j\left(\theta \right),$$

whence $R\stackrel{L}{=}{\Lambda }_a+I_{p}J$ and the assertion follows. □

4. Embedding the Lambert Law

In the last section, we tried to find a version of the stationary-excess condition (16) applicable to an arbitrary Poisson jump rate a while keeping the jump law of the Lambert BDLP fixed. Here, we fix the jump rate a and show that there is a scale family solution to the pair of in-law equations

$$X\stackrel{L}{=}{U}_a(X+J)\&X\stackrel{L}{=}{U}_a\mathcal{S}J,$$

(22)

where we recall that $U_a=U^{1/a}$; other notation and independence assumptions are as in Section 3, except that $L\left(X\right)$ and $L\left(J\right)$ will depend on a, a dependence we usually suppress in our notation.

Given $L\left(J\right)$ having a finite mean $\mu$, the perpetuity condition has a unique solution $L\left(X\right)$ for which the LST is

$$\chi \left(\theta \right)=\exp \left[-a{\int }_0^{\infty }\left(1-e^{-\theta x}\right)x^{-1}{\overline{F}}_J\left(x\right)dx\right],$$

(23)

where ${\overline{F}}_J\left(x\right)=P(J>x)$; see, e.g., p. 764 in [20] and references therein. Note that the finite mean condition is sufficient but not necessary. Our restriction to this condition is to ensure that the stationary-excess condition is not vacuous. It is clear that $L\left(X\right)$ is SD, and we will call it a perpetuity law. Thus, our problem is whether there is a law $L\left(J\right)$ such that the second condition in (22) holds. Theorem 5 below gives an affirmative answer, following from the preliminary results in Lemma 2, Theorem 4, and Remark 2.

Given a positive constant $\mu$, let $\mathcal{L}\left(\mu \right)=\left\{L\right(J):P(J\ge 0)=1,E(J)=\mu \}$. We also define the following parameter: if $0<a\ne 1$, then

$$q={\displaystyle \frac{a}{|1-a|}}.$$

(24)

Observe that

$$q/a=\left\{\begin{array}{cc}1+q\hfill & \mathrm{if}0<a<1,\hfill \\ q-1\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

Lemma 2. 

Fix positive constants $a\ne 1$ and μ. If there exists $L\left(J\right)$ in $\mathcal{L}\left(\mu \right)$ such that the two conditions of (22) are satisfied, then the LST χ of $L\left(X\right)$ solves the trinomial equation

$${\chi }^{-1/a}-{\chi }^{-1}=(1-a)\mu \theta .$$

(25)

In addition, $L\left(J\right)$ is infdiv and satisfies

$$P(J=0)=\left\{\begin{array}{cc}0\hfill & \mathit{if}a<1,\hfill \\ 1-a^{-1}\hfill & \mathit{if}a>1.\hfill \end{array}\right.$$

Proof. 

The right-hand sides of the identities (22) are equal in law and the factors $U_a$ can be cancelled, yielding $\mathcal{S}J\stackrel{L}{=}X+J$. Defining $j\left(\theta \right)=E\left[\exp \right(-\theta J\left)\right]$, it follows that $\chi \left(\theta \right)j\left(\theta \right)=(1-j(\theta \left)\right)/\left(\mu \theta \right)$, i.e.,

$$j\left(\theta \right)={\displaystyle \frac{1}{1+\mu \theta \chi \left(\theta \right)}}.$$

(26)

The stationary-excess condition yields the identity

$$\mu {\theta }^a\chi \left(\theta \right)=a{\int }_0^{\theta }(1-j\left(y\right))y^{a-2}dy.$$

Differentiating and eliminating $j\left(\theta \right)$ yields the first-order differential equation

$${\chi }^{\prime }\left(\theta \right)={\displaystyle \frac{-a\mu {\chi }^2\left(\theta \right)}{1+\mu \theta \chi \left(\theta \right)}}.$$

This can be recast as a linear first-order equation for $\theta$ as a function of $\chi$,

$${\displaystyle \frac{d\theta }{d\chi }}+{\displaystyle \frac{\theta }{a\chi }}={\displaystyle \frac{-1}{a\mu {\chi }^2}}.$$

Assuming now that $a\ne 1$, solving this subject to $\chi \left(0\right)=1$ yields (25).

Using (25) to eliminate $\mu \theta$ in (26) yields the general expression

$$j\left(\theta \right)={\displaystyle \frac{(1-a){\chi }^{-1}\left(\theta \right)}{{\chi }^{-1/a}\left(\theta \right)-a{\chi }^{-1}\left(\theta \right)}}.$$

(27)

If $a<1$, then

$$j\left(\theta \right)={\displaystyle \frac{(1-a){\chi }^{1/q}\left(\theta \right)}{1-a{\chi }^{1/q}\left(\theta \right)}}.$$

(28)

Let $({\mathcal{X}}_t:t\ge 0)$ denote the subordinator with ${\mathcal{X}}_1\stackrel{L}{=}X$. It follows from (28) that $J\stackrel{L}{=}{\sum }_{n=1}^N{Y}_n$, where $Y_n$ are independent copies of ${\mathcal{X}}_{1/q}$ and independent of N, which has the geometric law $P(N=n)=(1-a)a^{n-1}$ for $n\ge 1$. Hence, $j\left(\theta \right)$ is an infdiv LST; see, e.g., p. 84 in [1].

If $a>1$, then we can let $p=1/a$ and rearrange (27) to obtain

$$j\left(\theta \right)={\displaystyle \frac{1-p}{1-p{\chi }^{1/q}\left(\theta \right)}},$$

(29)

so $j\left(\theta \right)$ is again an infdiv LST.

We know from (23) that for arbitrary a, $L\left(X\right)$ is SD; hence, $\chi (\infty -)=0$. The assertion about $P(J=0)$ follows from (28) and (29). In particular, if $a>1$, then $P(J=0)=1-p$. □

Remark 1. 

We see from (25) that μ functions as a scale constant, so we can (and will, with no loss of generality) set $\mu =1$ in what follows; thus, $E\left(J\right)=1$.

We pause to remind the reader of Sato’s classification of infdiv laws on p. 65 of [21]. Here, we are concerned only with cases where there is no drift or Gaussian component and where, with reference to (1), the Lévy measure, say $\mathcal{L}$, is supported in $[0,\infty )$. We say that the corresponding law is of Type A if $\mathcal{L}\left(\right[0,1\left)\right)<\infty$, in which case the law is compound Poisson, and that it is of Type B if $\mathcal{L}\left(\right[0,1\left)\right)=\infty$ and ${\int }_0^{1}x\mathcal{L}\left(dx\right)<\infty$, in which case the infdiv law is supported on $[0,\infty )$ and assigns zero mass to $\left\{0\right\}$. The law is of Type C if ${\int }_0^{1}x\mathcal{L}\left(dx\right)=\infty$, in which case there is an intrinsic negative drift which makes the law two-sided. For Types A and B, almost all sample paths of the corresponding Lévy processes have bounded variation in compact intervals and have unbounded variation for Type C.

We know from the results in Section 3 that with $a=1$ (the Lambert case), $L\left(J\right)$ is of Type B and Lemma 2 directly extends this outcome to the case with $a\le 1$. The above discussion shows that despite the final assertion of Lemma 2, $L\left(J\right)$ can be chosen to be Type B for all a. The case with $a<1$ seems more fundamental in the sense that $L\left(X\right)$, as exhibited in Theorem 5 below, has a simpler form. The following corollary shows how the cases with $a<1$ and $a>1$ are related.

Corollary 1. 

Let $\chi (\theta ;a)$ denote the solution of (25) for $a\ne 1$. If $a>1$, then

$$\chi (\theta ;a)={\left(\chi (a\theta ;a^{-1})\right)}^a.$$

Proof. 

Replace a in (25) with $a^{-1}$ to obtain ${\left(\chi (\theta ;a^{-1})\right)}^{-a}-{\left(\chi (\theta ;a^{-1})\right)}^{-1}=(1-a^{-1})\mu \theta$ and then replace $\theta$ with $\alpha \theta$. The assertion follows. □

In order to show that the conditions in (22) are not vacuous, it remains to be shown that any solution of (25) satisfying $\chi \left(0\right)=1$ is the LST of a probability law. The constant $\mu$ functions as a scale parameter, and to lessen the notational burden we will, without loss of generality, set $\mu =1$ in (25). Below, we use the (formal) identity

$${\left.{\displaystyle \frac{d^{n-1}}{d{x}^{n-1}}}{x}^r\right|}_{x=1}=r(r-1)\cdots (r-n+2)={\displaystyle \frac{\Gamma (r+1)}{\Gamma (r-n+2)}}.$$

(30)

Theorem 4. 

Let $t>0$ and $0<a<1$ (so $q=a/(1-a)$); then, positive powers of the solution of (25) have a power series representation that converges for all complex-valued θ:

$${\chi }^t\left(\theta \right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{[-(1-a)\theta ]}^n}{n!}}·M\left(n;a,t\right),$$

(31)

where

$$M(s;a,t)={\displaystyle \frac{\Gamma (1+qt)}{\Gamma \left(qt\right)}}·{\displaystyle \frac{\Gamma (qt+qs/a)}{\Gamma (1+qt+qs)}}.$$

(32)

In addition, $M(s;a,t)$ is the bilateral LST of a spectrally positive Type C infdiv law, say $M(s;a,t)=E\left(e^{-Y(a,t)}\right)$. Finally, ${\chi }^t\left(s\right)$ is the LST of the infdiv random variable

$$X_{a,t}\stackrel{L}{=}(1-a)e^{-Y(a,t)}.$$

(33)

Proof. 

Defining $\zeta =1-{\chi }^{1/q}$, we can cast (25) (with $\mu =1$) into the canonical form $\zeta =\theta (1-a)f\left(\zeta \right)$, where $f\left(\zeta \right)={(1-\zeta )}^{1+q}$ is analytic in the open unit disc; in addition, ${\chi }^t={(1-\zeta )}^{qt}$. The Lagrange expansion (e.g., p. 145 in [22]) is applicable with the outcome

$${\chi }^t\left(\theta \right)=1-qt{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\left[(1-a)\theta \right]}^n}{n!}}{C}_n$$

where, using (30),

$$\begin{array}{ccc}\hfill C_n& =& {\left.{\displaystyle \frac{d^{n-1}}{d{\zeta }^{n-1}}}{(1-\zeta )}^{(q+1)n+qt-1}\right|}_{\zeta =0}={(-1)}^{(n-1)}{\displaystyle \frac{\Gamma \left(\right(q+1)n+qt)}{\Gamma \left(\right(q+1)n+qt-1-n+2)}}\hfill \\ & =& {(-1)}^{n-1}{\displaystyle \frac{\Gamma (qt+qn/a)}{\Gamma (1+qt+qn)}}.\hfill \end{array}$$

This establishes (31). It follows from the asymptotic equivalence $\Gamma (a+z)/\Gamma (b+z)\sim z^{a-b}$ as $z\to \infty$ (#5.11.12 in [3]) that $C_n=O\left(n^{-1}\right)$, and hence that ${\chi }^t\left(\theta \right)$ is an entire function.

The first gamma function quotient in (32) equals $qt$, and hence $M(n;a,t)=qt|C_n|$. We interpret the second gamma function quotient in (32) using Theorem 3.1 in [23], which asserts conditions under which such a quotient is the bilateral Laplace transform of a spectrally positive infdiv law and specifies an explicit expression for the corresponding Lévy density. This outcome is in essence a consequence of the integral representation of $\log \Gamma \left(z\right)$ derived by integrating the (Gauss) representation for the $\psi$-function, #5.9.12 in [3]. Comparing the algebraic form of (32) with (3.2) in [23], it follows that $M(s;a,t)$ is the bilateral LST of $L\left(Y\right(a,t\left)\right)$, as asserted, and that its Lévy density is

$${\ell }_{<1}\left(y\right)=y^{-1}\left[{\displaystyle \frac{e^{-aty}}{1-e^{-(1-a)y}}}-{\displaystyle \frac{e^{-(t+1/q)y}}{1-e^{-y/q}}}\right].$$

Moreover, the sequence $\left(M\right(n;a,t):n=0,1,\dots )$ comprises the elements of the moment sequence of the positive random variable $\exp (-Y(a,t\left)\right)$. The asymptotic estimate ${(\Gamma (A+qz))}^{1/z}\sim e^{-q}{\left(qz\right)}^q$ ($z\to \infty$) shows that ${\left(M(n;a,t)\right)}^{1/n}$ converges to a positive constant as $n\to \infty$, and hence that ${\chi }^t\left(\theta \right)$ is an entire function of $\theta$, meaning that this moment sequence is determinate. It follows for each $t>0$ that ${\chi }^t\left(\theta \right)$ is the LST of $L\left(X_{a,t}\right)$, as specified by (33). We conclude that $\chi \left(\theta \right)$ is the LST of a positive infdiv random variable $X_a=X_{a,1}$, and hence that $L\left(X_{a,t}\right)$ is the law of a subordinator at time t. □

Remark 2. 

If $a>1$, then $q=a/(a-1)$ and the above assertions hold with the following alterations. Defining $\zeta =1-{\chi }^{1/q}$, we have ${\chi }^t={(1-\zeta )}^{qt}$, and (25) becomes $\zeta =(a-1)\theta {(1-\zeta )}^q$. Lagrange reversion and arguing as above yields

$${\chi }^t\left(\theta \right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{[-(a-1)\theta ]}^n}{n!}}M(n;a,t),$$

where

$$M(s;a,t)={\displaystyle \frac{\Gamma (1+qt)}{\Gamma \left(qt\right)}}.{\displaystyle \frac{\Gamma (qt+qs)}{\Gamma (1+qt+qs/a)}}$$

is the bilateral LST of the Type-C infdiv random variable $Y(a,t)$, for which the Lévy density is

$${\ell }_{>1}\left(y\right)={\displaystyle \frac{1}{y}}·\left[{\displaystyle \frac{e^{-ty}}{1-e^{-ty/q}}}-{\displaystyle \frac{e^{-(a-1+at)y}}{1-e^{-(a-1)y}}}\right]={\ell }_{<1}(y:a^{-1},at).$$

The relation connecting ${\ell }_{<1}(·)$ and ${\ell }_{>1}(·)$ reflects Corollary 1. Finally, $X_{a,t}=(a-1)\exp (-Y(a,t))$.

The proof of Theorem 4 does not show whether or not $L\left(X_a\right)$ is SD. The next result shows that it is in fact a perpetuity law, and hence SD, as explained following (22). The proof is in essence a ‘reverse engineering’ of the proof of Lemma 2.

Theorem 5. 

If $0<a\ne 1$, then the law $L\left(X_a\right)$ of Theorem 4 and Remark 2 satisfies (22) with $L\left(J\right)$ having the LST (29). In particular, $L\left(X_a\right)$ is SD and $E\left(X_a\right)=a$.

Proof. 

Computing $1-j\left(\theta \right)$ from (26) yields

$$1-j\left(\theta \right)={\displaystyle \frac{1-{\chi }^{1/q}\left(\theta \right)}{1-a{\chi }^{1/q}\left(\theta \right)}}={\chi }^{1/a}\left(\theta \right)·{\displaystyle \frac{{\chi }^{-1/a}\left(\theta \right)-{\chi }^{-1}\left(\theta \right)}{1-a{\chi }^{1/q}\left(\theta \right)}}.$$

(34)

It follows from (25) that

$${\displaystyle \frac{1-j\left(\theta \right)}{\theta }}={\displaystyle \frac{1-a}{{\chi }^{-1/a}\left(\theta \right)-a{\chi }^{-1}\left(\theta \right)}}.$$

On the other hand, differentiating (25) yields

$$-{\displaystyle \frac{{\chi }^{\prime }\left(\theta \right)}{a\chi \left(\theta \right)}}={\displaystyle \frac{1-a}{{\chi }^{-1/a}\left(\theta \right)-a{\chi }^{-1}\left(\theta \right)}}={\displaystyle \frac{1-j\left(\theta \right)}{\theta }}.$$

(35)

Recalling that $E\left(J\right)=1$, this identity implies that $E\left(X_a\right)=a$.

Write the identity as $\theta {\chi }^{\prime }\left(\theta \right)+a\chi \left(\theta \right)=a\chi \left(\theta \right)j\left(\theta \right)$, i.e., ${\left({\theta }^a\chi \left(\theta \right)\right)}^{\prime }=a{\theta }^{a-1}\chi \left(\theta \right)j\left(\theta \right)$ and integrate to obtain the identity

$${\theta }^a\chi \left(\theta \right)=a{\int }_0^{\theta }\chi \left(y\right)j\left(y\right)y^{a-1}dy={\int }_0^{\theta }\chi \left(y\right)j\left(y\right)d{y}^a.$$

Substituting $y^a={\theta }^{a}u$ yields $\chi \left(\theta \right)={\int }_0^1\chi \left(\theta u^{1/a}\right)j\left(\theta u^{1/a}\right)du$, i.e.,

$$E\left(e^{-\theta X_a}\right)=E\left[e^{-\theta U_a(X_a+J)}\right],$$

the LST version of the perpetuity condition in (22).

Finally, it follows from (25) and (34), and $1/a=1+1/q$ that

$${\displaystyle \frac{1-j\left(\theta \right)}{\theta }}={\displaystyle \frac{(1-a){\chi }^{1/a}\left(\theta \right)}{1-a{\chi }^{1/q}\left(\theta \right)}}=\chi \left(\theta \right)j\left(\theta \right).$$

Hence, $\mathcal{S}J\stackrel{L}{=}{X}_a+J$, implying that $U_a\mathcal{S}J\stackrel{L}{=}{U}_a(X_a+J)\stackrel{L}{=}{X}_a$ and both conditions of (22) hold. □

Our final general result shows that $L\left(X_a\right)$ is a GGC for all $0<a\ne 1$. We need the following concepts. First, let $0<a\le 1$ and $L\left(S_a\right)$ denote the standard positive stable law with index a. Denote its density function by ${\sigma }_a\left(x\right)$. A known identity is $E\left(S_a^{-s}\right)=\Gamma (1+s/a)/\Gamma (1+s)$, which is finite if $s\ge 0$. Next, if $Z\ge 0$ is a random variable with distribution function $F\left(z\right)$ and moment function $\mathcal{M}\left(s\right)=E\left(Z^s\right)$ which is finite at least for $s\in [0,r]$, some $r>0$, then the order-r length biased version of Z is the random variable ${\mathcal{L}}_{r}Z$ for which the distribution function $d{F}_Z\left(z\right)\propto z^{r}dF\left(z\right)$ (equivalently, its moment function) is $\mathcal{M}(r+s)/\mathcal{M}\left(r\right)$.

Theorem 6. 

If $0<a\ne 1$, then $L\left(X_a\right)$ is a GGC.

Proof. 

The proof is in three parts. (a) Denote the CF of $L\left(X_a\right)$ by

$$\omega (\theta ;a)=-\log \chi (\theta ;a)={\int }_0^{\infty }\left(1-e^{-\theta x}\right)l(x,a)dx;$$

Theorem 4 implies that there is no drift term. It follows from Corollary 1 that if $a>1$, then $\omega (\theta ;a)=a\omega (a\theta ;a^{-1})$, i.e.,

$${\int }_0^{\infty }\left(1-e^{-\theta x}\right)l(x,a)dx=a{\int }_0^{\infty }\left(1-e^{-a\theta x}\right)l(x,a^{-1})dx.$$

It follows that $l(x;a)=l(x/a;a^{-1})$. Hence, if $xl(x;a)$ is completely monotone when $a<1$, then this also holds if $a>1$.

(b) Thus, it suffices to consider the case with $a<1$. Referring to the proof of Theorem 4, recall that we defined $\zeta =1-{\chi }^{1/q}$ so that $\zeta =\theta (1-a)f\left(\zeta \right)$, where $f\left(\zeta \right)={(1-\zeta )}^{1+q}$. We seek an expression for the CF (suppressing explicit a dependence), say $\omega \left(\theta \right)=-q^{-1}\log (1-\zeta )=g\left(\zeta \right)$. Hence, $g^{\prime }\left(\zeta \right)={\left(q(1-\zeta )\right)}^{-1}$. Lagrange reversion yields

$$\omega \left(\theta \right)=q^{-1}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\left[(1-a)\theta \right]}^n}{n!}}{\Delta }_n,$$

where

$${\Delta }_n={\left.{\displaystyle \frac{d^{n-1}}{d{\zeta }^{n-1}}}{(1-\zeta )}^{n(1+q)-1}\right|}_{\zeta =0}={(-1)}^{n-1}{\displaystyle \frac{\Gamma \left(\right(1+q\left)n\right)}{\Gamma \left(\right(1+q)n-1-n+2)}}={(-1)}^{n-1}{\displaystyle \frac{\Gamma (qn/a)}{\Gamma (1+qn)}},$$

where the last equality follows because $1+q=q/a$. Hence, term-wise differentiation, which is valid here, yields

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\theta \right)& =& {\int }_0^{\infty }{e}^{-\theta x}xl\left(x\right)dx={\displaystyle \frac{1-a}{q}}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{[-(1-a)\theta ]}^n}{n!}}{\displaystyle \frac{\Gamma \left(q\right(1+n)/a)}{\Gamma (1+q+qn)}}\hfill \\ & =& {\displaystyle \frac{1-a}{q^2}}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{[-(1-a)\theta ]}^n}{n!}}·{\displaystyle \frac{1}{n+1}}·{\displaystyle \frac{\Gamma \left(q\right(1+n)/a)}{\Gamma (q+qn)}}.\hfill \end{array}$$

Observing that $E\left(S_a^{-qn}\right)=\Gamma (qn/a)/a\Gamma \left(qn\right)$, we see that the last gamma function quotient is proportional to $E\left(T^n\right)$, where $T\stackrel{L}{=}{\mathcal{L}}_1\left(S_a^{-q}\right)$. Hence, ${\omega }^{\prime }\left(\theta \right)\propto E\left(e^{-\theta (1-a)Z}\right)$, where $Z\stackrel{L}{=}UT$.

Recalling that ${\sigma }_a\left(x\right)$ denotes the pdf of $S_a$, it follows that the pdf of $S_a^{-q}$ is $q^{-1}{x}^{-1-1/q}{\sigma }_a\left(x^{-1/q}\right)$, and hence that the pdf of T is $\propto x^{-1/q}{\sigma }_a\left(x^{-1/q}\right)$. We conclude that

$$xl\left(x\right)=K{\int }_x^{\infty }{y}^{-1-1/q}{\sigma }_a\left(y^{-1/q}\right)dy.$$

(c) Observe that the characteristic function of $S_a$ is

$$E\left(e^{i\tau S_a}\right)=e^{-{(-i\tau )}^a}=\exp \left[-{\left|\tau \right|}^a{e}^{(i\pi a/2)sign\left(\tau \right)}\right].$$

Comparing this with (4.4.14) in [24] and then with (4.4.12) there, we can note that for us the parameters $\beta =1$ and $\Phi =\pi a/2$ in [24] will lead to the following integral representation: if $x>0$, then

$${\sigma }_a\left(x\right)={\pi }^{-1}q{x}^{-1-q}{\int }_{-\pi /2}^{\pi /2}{e}^{-x^{-q}U\left(v\right)}U\left(v\right)dv,$$

where

$$U\left(v\right)={\left[{\displaystyle \frac{\sin \left(a\right(v+\pi /2\left)\right)}{\cos v}}\right]}^q·{\displaystyle \frac{\cos (\pi a/2-(1-a\left)v\right)}{\cos v}}$$

and $U\left(v\right)>0$ if $-\pi /2<v<\pi /2$. Consequently,

$${\sigma }_a\left(y^{-1/q}\right)={\pi }^{-1}q{y}^{1+1/q}{\int }_{-\pi /2}^{\pi /2}{e}^{-yU\left(v\right)}U\left(v\right)dv,$$

i.e., the function $y^{-1-1/q}{\sigma }_a\left(y^{-1/q}\right)$ is CM. It follows that $xl\left(x\right)$ is CM, and hence that $C\left(\theta \right)\in \mathcal{T}\mathcal{B}$. The assertion follows. □

The above results provide another proof that the Lambert W is a Thorin–Bernstein function, as follows.

Corollary 2. 

(a) The limit

$$\chi (\theta ;1):=\underset{a\to 1}{\lim }\chi (\theta ;a)=W\left(\theta \right)/\theta =\exp (-W\left(\theta \right)$$

is the LST of a GGC law.

(b) The moment function of the Lambert subordinator is

$$E\left({\Lambda }_t^s\right)=t{(t+s)}^{s-1}.$$

Proof. 

(a) Let $a<1$, write (25) (with $\mu =1$) as

$${\displaystyle \frac{\chi (\theta ;a)-{\chi }^{1/a}(\theta ;a)}{1-a}}=\theta {\chi }^{1+1/a}(\theta ;a),$$

and let $a\to 1$, evaluating the left-hand side using L’Hopitâl’s rule. The result is

$$-{\left.{\displaystyle \frac{\partial \chi (\theta ;a)}{\partial a}}\right|}_{a=1}+a^{-1}{\left.{\displaystyle \frac{\partial \chi (\theta ;a)}{\partial a}}\right|}_{a=1}-\chi (\theta ;1)\log \chi (\theta ;1)=\theta {\chi }^2(\theta ;1),$$

i.e., $\log \chi (\theta ;1)=-\theta \chi (\theta ;1)$. Set $w\left(\theta \right)=\chi (\theta ;1)$ to obtain $\exp (-w(\theta \left)\right)=w\left(\theta \right)/\theta$, which using (2) implies that $w\left(\theta \right)=W\left(\theta \right)$. However, $\chi (\theta ;1)$ is the limiting LST of a family of GGC laws; hence, $L\left(X_1\right)$ is itself a GGC.

(b) Taking the case $a<1$, it follows from (31) and (32), the gamma function functional equation, and $q+1=q/a$ that

$$E\left(X_{a,t}^s\right)={(1-a)}^s{\displaystyle \frac{t}{t+s}}{\displaystyle \frac{\Gamma \left(q\right(t+s)+s)}{\Gamma \left(q\right(t+s\left)\right)}}.$$

The gamma function quotient $\sim {\left[q(t+s)\right]}^s$ as $a\uparrow 1$; thus, it follows that $E\left(X_{a,t}\right)\to t{(t+s)}^{s-1}$. On the other hand, Assertion (a) implies that $X_{a,t}\stackrel{L}{\to }{\Lambda }_t$; hence, Assertion (b) follows from a moment function convergence theorem, e.g., Theorem 30.2 in [25]. □

The following result asserts the asymptotic form of $\chi \left(\theta \right)$ as $\theta \to \infty$, which in turn implies the left-hand tail behaviour of $L\left(X\right)$.

Lemma 3. 

As $\theta \to \infty$,

$$\chi (\theta ;a)\sim \left\{\begin{array}{cc}{\left[(1-a)\theta \right]}^{-a}\hfill & ifa<1,\hfill \\ {\left[(a-1)\theta \right]}^{-1}\hfill & ifa>1.\hfill \end{array}\right.$$

Hence, as $x\to 0+$,

$$P(X_a\le x)\sim \left\{\begin{array}{cc}{\displaystyle \frac{{(1-a)}^{-a}{x}^a}{\Gamma (1+a)}}\hfill & \mathrm{if}a<1,\hfill \\ {\displaystyle \frac{x}{a-1}}\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

Proof. 

It is evident from (25) that $\chi \left(\theta \right)\to 0$ as $\theta \to \infty$, and the convergence rates easily follow. The left-hand tail estimates follow from a Tauberian theorem, e.g., p. 38 in [18]. □

It follows from these asymptotic estimates and Theorem 1 in [26] that $L\left(X_a\right)$ is the limiting law of a shot noise process. Define the stochastic process $(Y_t:t\ge 0)$, where

$$Y_t=\sum _{{\tau }_i\le t}{\Pi }_i{e}^{-\kappa (t-{\tau }_i)},$$

in which the noise impulses ${\Pi }_i$ are independent copies of $\Pi >0$, the ${\tau }_i$ are the jump times of an independent Poisson process having rate r, and $\kappa$ is a positive constant. The notation in [26] differs from ours; Theorem 1 in this reference implies that if $\left(Y_t\right)$ has the limit law $L\left(X_a\right)$, then $r/\kappa =a\wedge 1$ and $\pi \left(\theta \right):=E\left(e^{-\theta \Pi }\right)=1-(r/\kappa )\theta {\omega }^{\prime }(\theta ;a)$.

It follows immediately from (35) that if $0<a<1$, then $\pi \left(\theta \right)=j\left(\theta \right)$. If $a>1$, then using (25) to eliminate $1-a$ in (35) leads after some algebra to

$$\pi \left(\theta \right)={\displaystyle \frac{(1-p){\chi }^{1/q}\left(\theta \right)}{1-p{\chi }^{1/q}\left(\theta \right)}}=j\left(\theta \right){\chi }^{1/q}\left(\theta \right),$$

the second equality following from (29). If we set $\kappa =1$, then $\left(Y_t\right)$ is the BDLP of the SD law $L\left(X_a\right)$, i.e., a compound Poisson process for which the jump rate is $a\wedge 1$ and a jump law for which the LST is $\pi \left(\theta \right)$; see (3) in [26]. This outcome can also be inferred from identities derived above.

The case $a=1$ corresponds to the Lambert law; then, ${\chi }_1\left(\theta \right)=W\left(\theta \right)/\theta \sim \left(\log \theta \right)/\theta$ as $\theta \to \infty$, and it follows that

$$P(\Lambda \le x)\sim x\log \left(x^{-1}\right),(x\to 0+).$$

It follows too from (9) that

$$\pi \left(\theta \right)={\left(1+W\left(\theta \right)\right)}^{-1}=E\left(e^{-\theta {\Lambda }_{ϵ}}\right),$$

i.e., $L(\Pi )$ is the law of the Lambert subordinator stopped at a random independent time having the unit exponential law.

Another consequence from Lemma 3 is that the density function $f(x;a)$ of $L\left(X_a\right)$ is non-increasing in $(0,\infty )$. The SD nature of $L\left(X_a\right)$ implies the existence of $f(x;a)$ and that it is unimodal. Moreover, $f(x;1)$, the Lambert case, is completely monotone (Theorem 3.2 in [12]). For $a<1$, it follows from the distribution function estimate in Lemma 3 that $f(0+;a)=\infty$, the mode is at the origin. In general, the so-called canonical density $k(x;a):=x\ell (x;a)$ of $L\left(X_a\right)$ is non-increasing, a fundamental SD property. Its Laplace-transform is

$$\widehat{k}\left(\theta \right)=-{\displaystyle \frac{{\chi }^{\prime }(\theta ;a)}{\chi (\theta ;a)}}={\displaystyle \frac{a\chi (\theta ;a)}{1+\theta \chi (\theta ;a)}},$$

where the final evaluation is a rearrangement of the evaluation of ${\chi }^{\prime }(\theta ;a)$ in the proof of Lemma 2. For $a>1$, it follows from Lemma 3 that $\widehat{k}(\theta ;a)\sim {\theta }^{-1}$ as $\theta \to \infty$, and hence that $k(0+;a)=1$, invoking Theorem 1.7.2b in [18]. T together with Theorem 2.17 on p. 236 in [1] or Theorem 53.4(ii) in [21], this outcome implies that $f(x;a)$ is non-increasing. Summarising, if $a>0$, then $f(x;a)$ has its unique mode at the origin.

We turn now to random variable representations. The representation $X_{a,t}\stackrel{L}{=}|a-1|e^{-Y(a,t)}$ inherent in Theorem 4 and Remark 2 is one of many possibilities. It provides tidier versions of the above Lévy densities that emerge by applying the basic gamma function relation to the quotients in (32) and Remark 2 to obtain

$$E\left(X_{a,t}^s\right)=\left\{\begin{array}{cc}{(1-a)}^s·{\displaystyle \frac{t}{t+s}}·{\displaystyle \frac{\Gamma (qt+qs/a)}{\Gamma (qt+qs)}}\hfill & \mathrm{if}a<1,\hfill \\ {(a-1)}^s·{\displaystyle \frac{at}{at+s}}·{\displaystyle \frac{\Gamma (qt+qs)}{\Gamma (qt+qs/a)}}\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

(36)

Here, the gamma function quotients are again the bilateral Laplace transforms of a Type-C infdiv random variable $\tilde{Y}(a,t)$. As above, it follows that

$$X_{a,t}\stackrel{L}{=}\left\{\begin{array}{cc}(1-a)U^{1/t}{e}^{-\tilde{Y}(a,t)}\hfill & \mathrm{if}0<a<1,\hfill \\ (a-1)U^{1/at}{e}^{-\tilde{Y}(a,t)}\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

In addition, from Theorem 3.1 in [23], the Lévy density of $\tilde{Y}(a,t)$ is

$$\tilde{\ell }\left(y\right)=\left\{\begin{array}{cc}{y}^{-1}\left({\displaystyle \frac{e^{-aty}}{1-e^{-(1-a)y}}}-{\displaystyle \frac{e^{-ty}}{1-e^{-y/q}}}\right)\hfill & \mathrm{if}a<1,\hfill \\ y^{-1}\left({\displaystyle \frac{e^{-ty}}{1-e^{-y/q}}}-{\displaystyle \frac{e^{-aty}}{1-e^{-(a-1)y}}}\right)\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

Lemma 3.1 in [23] yields a representation of $Y(a,t)$ and $\tilde{Y}(a,t)$ as infinite series of independent random terms. The nth term is a weighted and centred version of the product $I_n{ϵ}_n$, where $I_n$ has a Bernoulli law depending on n, ${ϵ}_n$ have unit exponential laws, and these factors are independent; hence, $X_{a,t}$ has representations as an infinite product of independent factors.

A quite different representation involves a length-biased version of negative powers of the positive stable random variable $S_a$, where $a<1$. Recall that $E\left(S_a^{-s}\right)=a^{-1}\Gamma (s/a)/\Gamma \left(s\right)$ and let $L\left(B_{c,d}\right)$ denote the beta law with parameters c and d. Note too that $E\left(U^{s/t}\right)=t/(s+t)$. Our representation follows from (36) and Lemma 2.1 in [23], which when read in reverse asserts a random variable representation for moment functions expressed as gamma function quotients having the form in (36).

Lemma 4. 

If $0<a\ne 1$, then

$$X_{a,t}\stackrel{L}{=}\left\{\begin{array}{cc}(1-a)U^{1/t}{\left(B_{qt,t}\right)}^{1+q}{\mathcal{L}}_t{S}_a^{-q}\hfill & \mathrm{if}0<a<1,\hfill \\ (a-1)U^{1/at}{\left(B_{qt,at}\right)}^q{\mathcal{L}}_{at}{S}_{1/a}^{-(q-1)}\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

(37)

These representations substantially simplify in the case with $t=q^{-1}$, in which case it follows from (36) that

$$X_{a,q^{-1}}\stackrel{L}{=}\left\{\begin{array}{cc}(1-a){\left(U{S}_a^{-1}\right)}^q\hfill & \mathrm{if}0<a<1,\hfill \\ (a-1){\left(U{S}_{1/a}^{-1}\right)}^{q/a}\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

(38)

In particular, if $a>1$, then $X_{a,q}\stackrel{L}{=}a{X}_{1/a,\overline{q}}$, where $\overline{q}={(a-1)}^{-1}=a^{-1}/(1-a^{-1})$, which is another manifestation of Corollary 1.

We next look at some examples.

Example 1. 

If $a=\frac{1}{2}$, then (25) is a quadratic equation to which the relevant solution is

$$\chi (\theta ;\frac{1}{2})={\displaystyle \frac{2}{1+\sqrt{1+2\theta }}}.$$

If ε has the standard exponential law, then

$$\xi \left(\theta \right):=E\left(e^{-2\theta {\epsilon }^2{S}_{1/2}}\right)=E\left(e^{-\epsilon \sqrt{2\theta }}\right)={\left(1+\sqrt{2\theta }\right)}^{-1}.$$

Though not relevant here, this is the LST of a positive Linnik law.

If Z is a positive random variable with LST $\zeta \left(\theta \right)$, then we define the exponential-tilt operator ${\tau }_{\rho }$ such that

$$P({\tau }_{\rho }\left(Z\right)\in dx)={\displaystyle \frac{e^{-\rho x}P(Z\in dx)}{\zeta \left(\rho \right)}},$$

provided that the denominator is finite. If $\rho >0$, this is the case; then,

$$E\left(e^{-\theta {\tau }_{\rho }\left(Z\right)}\right)={\displaystyle \frac{\zeta (\rho +\theta )}{\zeta \left(\rho \right)}}.$$

Observing that $\chi (\theta ;\frac{1}{2})=\xi (\frac{1}{2}+\theta )/\xi \left(\frac{1}{2}\right)$, we have the representation

$$X_{1/2}\stackrel{L}{=}{\tau }_{1/2}\left({\epsilon }^2{S}_{1/2}\right).$$

On the other hand, referring to Lemma 3, we have the equivalent representation

$$X_{1/2}\stackrel{L}{=}\frac{1}{2}U{S}_{1/2}^{-1}.$$

This representation yields evaluation of the density function as

$$f(x;\frac{1}{2})={\left(\pi x\right)}^{-\frac{1}{2}}{\int }_1^{\infty }{v}^{-3/2}{e}^{-xv}dv,$$

a completely monotone function.

Finally, noting that

$$\chi (\theta ;\frac{1}{2})={\theta }^{-1}\left[\sqrt{1+2\theta }-1\right],$$

binomial expansion leads to the moment identity

$$E\left(X_{1/2}^n\right)={\displaystyle \frac{2^n\Gamma (n+\frac{1}{2})}{(n+1)\Gamma \left(\frac{1}{2}\right)}}.$$

On the other hand, it follows from Theorem 4 that

$$E\left(X_{1/2}^n\right)=2^{-n}{\displaystyle \frac{\Gamma (1+2n)}{\Gamma (2+n)}}=2^{-n+1}{\displaystyle \frac{\Gamma \left(2n\right)}{(n+1)\Gamma \left(n\right)}}.$$

Equating these expressions yields the identity

$$\Gamma \left(\frac{1}{2}\right)\Gamma \left(2n\right)=2^{2n-1}\Gamma \left(n\right)\Gamma (n+\frac{1}{2}),$$

which in essence is Legendre’s duplication formula.

Example 2. 

If $a=1/3$, then $q=\frac{1}{2}$ and (25) can be written as the reduced cubic equation $y^3-y-2\theta /3=0$, where $y=1/\chi$ and $\chi \left(\theta \right)=\chi (\theta ;1/3)$. The cubic form $y^3-y$ has zeros at $y=-1,0 and 1,$ and is strictly increasing and positive if $y>1$. It follows that the relevant solution of the cubic equation lies along this positive limb. The discriminant of the cubic equation is ${\Delta }^2=({\theta }^2-1/3)/9$ and it is positive if ${\theta }^2>1/3$. In this case, the Cardano formula (e.g., p. 118 in [27]) gives the solution for χ as

$$\chi \left(\theta \right)=3^{1/3}{\left({[\theta +\sqrt{{\theta }^2-1/3}]}^{1/3}+{[\theta -\sqrt{{\theta }^2-1/3}]}^{1/3}\right)}^{-1}.$$

If ${\Delta }^2<0$, the trigonometrical solution algorithm (p. 121 in [27]) specifies numbers $\rho >0$ and ϕ defined by the relations $\rho \cos \varphi =\theta /3$ and $\rho \sin \varphi ={(1/3-{\theta }^2)}^{1/2}/3$. It follows that $\rho ={\left(3\sqrt{3}\right)}^{-1}$ and that the general solution is $y=2{\rho }^{1/3}\cos ((\theta +2k\pi )/3)$, where $k=0,1, or 2$. The solution satisfying $\chi \left(0\right)=1$ is that with $k=0$, i.e.,

$$\chi (\theta ;1/3)={\displaystyle \frac{3\sqrt{3}}{2}}\sec \left({\displaystyle \frac{\varphi }{3}}\right),(0\le \theta \le 3^{-1/2}).$$

These expressions for $\chi \left(\theta \right)$ are difficult to interpret. However, it follows from (38) that

$$X_{1/3,2}\stackrel{L}{=}{\displaystyle \frac{2}{3}}\sqrt{U{S}_{1/3}^{-1}}.$$

Recall that $E\left(S_{1/3}^{-s}\right)=\Gamma (1+3s)/\Gamma (1+s)$ and observe that taking $\alpha =1+3s$ in #9.10.17 of [3] gives the Mellin transform identity $3{\int }_0^{\infty }{z}^{3s}\mathrm{Ai}\left(z\right)dz=3^{-s}\Gamma (1+3s)/\Gamma (1+s)$, where $\mathrm{Ai}\left(z\right)$ is the Airy function. Hence, $V:={\left(3S_{1/3}^{-1}\right)}^{-1}$ has the density function

$$f_V\left(v\right)=v^{-2/3}Ai\left(v^{1/3}\right),(v>0).$$

It follows that the density function of $X_{1/3,2}$ is

$$f(x;1/3,2)={(3x/128)}^{1/3}{\int }_1^{\infty }{y}^{-5/3}Ai\left({\left({\displaystyle \frac{3y{x}^2}{4}}\right)}^{1/3}\right)dy.$$

Similar results can be derived if $a=3$ or, from (38),

$$X_{3,2/3}\stackrel{L}{=}2{\left(U{S}_{1/3}^{-1}\right)}^{1/2}\stackrel{L}{=}3X_{1/3,2}.$$

Example 3. 

If $a=2$, then solving (25) yields

$$\chi (\theta ;2)={\left({\displaystyle \frac{2}{1+\sqrt{1+4\theta }}}\right)}^2,$$

implying that

$$X_2\stackrel{L}{=}2\left(X_{1/2}^{\prime }+X_{1/2}^{\prime \prime }\right),$$

where the summands are independent copies of $X_{1/2}$. This illustrates Corollary 1.

Here, $q=2$; taking $t=\frac{1}{2}$, (37) yields

$$X_{2,1/2}\stackrel{L}{=}U{S}_{1/2}^{-1}.$$

Thus, $X_2$ is the sum of independent copies of $X_{2,1/2}$, and we can obtain a representation for this sum from Corollary 2 as follows. Taking $t=1$ there, we see that the gamma quotient is

$${\displaystyle \frac{\Gamma (2+2n)}{\Gamma (2+n)}}={\displaystyle \frac{1}{1+n}}·{\displaystyle \frac{\Gamma (3+2n)}{\Gamma \left(3\right)\Gamma (2+n)}}.$$

The right-hand side gamma function quotient is $E{\left({\mathcal{L}}_1{S}_{1/2}^{-1}\right)}^n$, and hence

$$X_2\stackrel{L}{=}{U}^{\prime }{U}^{\prime \prime \frac{1}{2}}{\mathcal{L}}_1\left(S_{1/2}^{-1}\right),$$

where the primed Us are independent copies of U.

5. BDLP Ladders: Examples

Let ${\omega }_0$ be the Lévy density of a positive infdiv law $L\left(X_0\right)$, i.e.,

$${\omega }_0\left(x\right)=0\mathrm{if}x<0,\mathrm{and}{\int }_0^{\infty }(x\wedge 1){\omega }_0\left(x\right)dx<\infty .$$

Let $C_0\left(\theta \right)={\int }_0^{\infty }(1-e^{-\theta x}){\omega }_0\left(x\right)dx$ denote the corresponding CF and let $(S_{0,t}:t\ge 0)$ be the drift-free subordinator determined by ${\omega }_0$, i.e., $S_{0,1}\stackrel{L}{=}{X}_0$. If ${\int }_1^{\infty }\left(\log x\right){\omega }_0\left(x\right)dx<\infty$, then the stochastic integral

$$X_1\stackrel{L}{=}{\int }_0^{\infty }{e}^{-t}d{S}_{0,t}$$

exists and $L\left(X_1\right)$ is SD, i.e., $\left(S_{0,t}\right)$ is the BDLP for $L\left(X_1\right)$. The Lévy density and CF of $L\left(X_1\right)$ are

$$C_1\left(\theta \right)={\int }_0^{\theta }{y}^{-1}{C}_0\left(y\right)dy\mathrm{and}{\omega }_1\left(x\right)=x^{-1}{\int }_x^{\infty }{\omega }_0\left(y\right)dy,$$

respectively, and $\left(S_{1,t}\right)$ denotes the drift-free subordinator determined by $L\left(X_1\right)$. The first identity is the known relation between the CF of an SD law and that of its BDLP. The second identity follows from the first.

We assume in the sequel that ${\omega }_0$ satisfies the log-condition, i.e., that ${\int }_1^{\infty }{\left(\log x\right)}^n{\omega }_0\left(x\right)dx<\infty$ for all positive-integer n. For each n, the function ${\left(\log (1+x)\right)}^n$ is submultiplicative (p. 159 in [21]); hence, the condition $E\left({\left(\log (1+X_0)\right)}^n\right)<\infty$ for all n is equivalent to the log-condition for ${\omega }_0\left(x\right)$. Clearly, ${\omega }_0$ satisfies the log-condition if ${\int }_1^{\infty }{x}^{\delta }{\omega }_0\left(x\right)dx<\infty$ for some $\delta >0$. The examples below satisfy this condition.

Iterating the construction of $L\left(X_1\right)$ generates a sequence of SD laws $\{L\left(X_n\right):n=1,2,\dots \}$ related by the BDLP representation

$$X_{n+1}\stackrel{L}{=}{\int }_0^{\infty }{e}^{-t}d{S}_{n,t},$$

(39)

where $\left(S_{n,t}\right)$ is a drift-free subordinator with Lévy density ${\omega }_n$. It generates the Lévy density

$${\omega }_{n+1}\left(x\right)=x^{-1}{\int }_x^{\infty }{\omega }_n\left(y\right)dy$$

(40)

and the CF

$$C_{n+1}\left(\theta \right)={\int }_0^{\theta }{y}^{-1}{C}_n\left(y\right)dy.$$

(41)

We call this construction a BDLP ladder to emphasise the focus on the sequence of laws $L\left(X_n\right)$ derived from an initial positive infdiv law $L\left(X_0\right)$. Note that ${\omega }_0$ can be replaced with an arbitrary Lévy measure (on $[0,\infty )$); however, our examples satisfy the more restrictive context, and there is no essential loss of generality. In this section we consider three examples, starting with a very simple one.

Example 4 

(The stable ladder). Suppose that $X_0\stackrel{L}{=}{S}_{0,1}\sim \mathrm{stable}(\alpha ,K)$, meaning that ${\omega }_0\left(x\right)=K{x}^{-\alpha -1}$ with $0<\alpha \le 1$ and $K>0$. A moment’s reflection shows that ${\omega }_n\left(x\right)={\alpha }^{-n}K{x}^{-\alpha -1}$, i.e., $X_n\sim \mathrm{stable}(\alpha ,{\alpha }^{-n}K)$. This ladder extends to $n<0$. Also, $C_n\left(\theta \right)\propto {\alpha }^{-n}{\theta }^{\alpha }$, and ${\alpha }^{n/\alpha }{X}_n\sim \mathrm{stable}(\alpha ,K)$.

Example 5 

(The gamma ladder). Let $a>0$ and ${\omega }_0\left(x\right)=a{e}^{-x}$, i.e., $\left(S_{0,t}\right)$ is a compound Poisson process with rate a and a unit exponential jump law. Here, $C_0\left(\theta \right)=a\theta /(1+\theta )$, so $C_1\left(\theta \right)=a\log (1+\theta )$ and $L\left(X_1\right)$ is the standard gamma $\left(a\right)$ law, a widely known result.

Assuming that $0<\theta <1$ and expressing $\log (1+\theta )$ as a power series, repeated use of (41) yields

$$C_n\left(\theta \right)=-a{\mathrm{Li}}_n(-\theta )=-a{\displaystyle \sum _{j=1}^{\infty }}{j}^{-n}{(-\theta )}^j,$$

(42)

where ${\mathrm{Li}}_n(·)$ is a polylogarithm function, #25.12.10 in [3]. The integral representation, #25.12.11 in [3], yields

$$C_n\left(\theta \right)={\displaystyle \frac{a\theta }{\Gamma \left(n\right)}}{\int }_0^{\infty }{\displaystyle \frac{y^{n-1}}{e^y+\theta }}dy.$$

(43)

However,

$${\displaystyle \frac{\theta }{e^y+\theta }}=1-{\displaystyle \frac{e^y}{e^y+\theta }}=e^y{\int }_0^{\infty }\left[\exp (-x{e}^y)-\exp (-x(\theta +e^y))\right]dx,$$

whence

$${\omega }_n\left(x\right)={\displaystyle \frac{a}{\Gamma \left(n\right)}}{\int }_0^{\infty }{y}^{n-1}\exp \left(-x{e}^y\right)e^{y}dy={\displaystyle \frac{a}{\Gamma \left(n\right)}}{\int }_1^{\infty }{e}^{-xz}{\left(\log z\right)}^{n-1}dz.$$

If $n\ge 2$, then integration by parts yields

$$x{\omega }_n\left(x\right)={\displaystyle \frac{a}{\Gamma (n-1)}}{\int }_1^{\infty }{e}^{-xz}{\left(\log z\right)}^{n-2}{z}^{-1}dz.$$

Thus, $x{\omega }_n\left(x\right)$ is completely monotone, so $C_n\left(\theta \right)$ is a Thorin–Bernstein function (p. 84 in [28]) and $L\left(X_n\right)$ is a generalized gamma convolution.

It follows from (43) that $C_n^{\prime }\left(0\right)=a$ for all $n\ge 0$, i.e., $E\left(X_n\right)=a$, and from (42) that $C_n\left(\theta \right)\to a\theta$ as $n\to \infty$, whence $X_n\stackrel{L}{\to }a$. Note that

$$C_n\left(\theta \right)-a\theta =-2^{-n}a{\theta }^2(1+o\left(1\right)),(n\to \infty )$$

yields the refinement

$$2^{n/2}\left(X_n-a\right)\stackrel{L}{\to }\sqrt{2a}\mathcal{N},$$

where $\mathcal{N}$ has the standard normal law. We will see in the next section that this outcome holds for all BDLP ladders under a mild additional moment condition.

Example 6 

(The Lambert ladder). Here, we take $\left(S_{0,t}\right)$ as the compound Poisson BDLP in (5). From (6), ${\omega }_0\left(x\right)={\int }_0^{\infty }s{e}^{-xs}\nu \left(ds\right)$, where ν is the probability measure having the Stieltjes transform

$$W^{\prime }\left(\theta \right)={\int }_0^{\infty }{\displaystyle \frac{\nu \left(ds\right)}{\theta +s}}.$$

It follows that ${\int }_0^{\infty }x{\omega }_0\left(x\right)dx=1$, and from (40) that ${\int }_0^{\infty }x{\omega }_n\left(x\right)dx=1$. Since $C_1\left(\theta \right)=W\left(\theta \right)$, we have from (9) that

$$C_0\left(\theta \right)=\theta W^{\prime }\left(\theta \right)={\displaystyle \frac{W\left(\theta \right)}{1+W\left(\theta \right)}}.$$

We next show that if $n\ge 1$, then $C_n\left(\theta \right)$ is a polynomial function of $W\left(\theta \right)$.

Theorem 7. 

If $n\ge 1$, then the CFs of the Lambert ladder are given by

$$C_n\left(\theta \right)=W\left(\theta \right)Q_{n-1}\left(W\left(\theta \right)\right),$$

(44)

where $Q_n\left(w\right)$ is a polynomial of degree n satisfying $Q_n\left(0\right)=1$ and

$$w{Q}_n\left(w\right)={\int }_0^w(1+v)Q_{n-1}\left(v\right)dv.$$

(45)

Proof. 

If $n=1$, then (44) holds with $Q_0\left(\theta \right)=1$. Assuming that (44) holds for some n, it follows from (41) that

$$\begin{array}{ccc}\hfill C_{n+1}\left(\theta \right)& =& {\int }_0^{\theta }{y}^{-1}W\left(y\right)Q_{n-1}\left(W\left(y\right)\right)dy={\int }_0^{\theta }(1+W\left(y\right))W^{\prime }\left(y\right)Q_{n-1}\left(W\left(y\right)\right)dy\hfill \\ & =& {\int }_0^{W\left(\theta \right)}(1+v)Q_{n-1}\left(v\right)dv=W\left(\theta \right)Q_n\left(W\left(\theta \right)\right).\hfill \end{array}$$

It is evident that the recursion (45) specifies a sequence of polynomials; hence, the assertion follows by induction. □

Computing recursively with (45) yields the first few polynomials

$$Q_1\left(w\right)=1+\frac{1}{2}w,Q_2\left(w\right)=1+{\displaystyle \frac{3}{4}}w+{\displaystyle \frac{1}{6}}{w}^2$$

and

$$Q_3\left(w\right)=1+{\displaystyle \frac{7}{8}}w+{\displaystyle \frac{11}{36}}{w}^2+{\displaystyle \frac{1}{24}}{w}^3.$$

It follows from (45) that the coefficients of $Q_n$ are positive, and hence that these polynomials are convex increasing in $[0,\infty )$. In addition, $Q_n\left(0\right)=1$, $Q_2\left(w\right)>Q_1\left(w\right)$ if $w>0$; hence, by induction from (45), $Q_{n+1}\left(w\right)>Q_n\left(w\right)$ if $w>0$.

Differentiating (45) yields

$$Q_n\left(w\right)+w{Q}_n^{\prime }\left(w\right)=(1+w)Q_{n-1}\left(w\right),$$

(46)

from which we see that the formally defined generating function $Q(w,z)={\sum }_{n\ge 0}{Q}_n\left(w\right)z^n$ satisfies

$${\displaystyle \frac{\partial Q}{\partial w}}+(1-(1+w)z)Q=1.$$

Solving this shows that the series defining Q converges if $\left|z\right|<1$ and that

$$\begin{array}{ccc}\hfill Q(w,z)& =& {\left(wz\right)}^{-(1-z)}\gamma (1-z,wz)\hfill \\ & =& {(1-z)}^{-1}{e}^{wz}M(1-z,2-z,-wz)\hfill \\ & =& {(1-z)}^{-1}M(1,2-z,wz),\hfill \end{array}$$

(47)

where $\gamma (a,z)={\int }_0^z{x}^{a-1}{e}^{-x}dx$ is the incomplete gamma function and $M(·,·,·)$ is the Kummer function; see #13.6.5 in [3] for these connections.

Writing $Q_n\left(z\right)={\sum }_{j=0}^n{A}_{nj}{w}^j$, we have $A_{n0}=1$; from (46), we have $(1+n)A_{nn}=A_{n-1,n-1}$, i.e., $A_{nn}=1/(n+1)!$. Explicit expressions for $A_{nj}$ can in principle be obtained from (47), because extracting the coefficient of $w^j$ yields

$${\displaystyle \sum _{n=0}^{\infty }}{A}_{n+j,j}{z}^n={\displaystyle \prod _{i=1}^{j+1}}{(i-z)}^{-1}.$$

Differentiation yields

$$A_{n,n-1}={\displaystyle \frac{H_n}{n!}}\mathrm{and}{A}_{n,n-2}={\displaystyle \frac{H_{n-1}^2+H_{n-1}^{\left(1\right)}}{2(n-1)!}},$$

(48)

where $H_n$ is the n-th harmonic number and $H_n^{\left(r\right)}={\sum }_{i=1}^n{i}^{-r-1}$ (so $H_n=H_n^{\left(0\right)}$). Partial fraction expansions yield $A_{n1}=1-2^{-n}$ if $n\ge 1$ and $A_{n2}=\frac{1}{2}(1-2^{-(n-2)}+3^{-n+1})$ if $n\ge 2$. The evaluations in (48) can be used to show that the $Q_n$ do not satisfy a three-term recurrence $w{Q}_n={\alpha }_n{Q}_{n+1}-({\alpha }_n+{\beta }_n)Q_n+{\beta }_n{Q}_{n-1}$. Consequently, there is no orthogonality measure for these polynomials.

We conclude by emphasising that the polynomial forms $W\left(\theta \right)Q_n\left(W\left(\theta \right)\right)$ are Bernstein functions. The above examples illustrate the general proposition in [11] that if $L\left(X_0\right)$ is a generalised gamma convolution, then so is each $L\left(X_n\right)$.

6. BDLP Ladders: Some Theory

Let ${\mathcal{W}}_0$ denote the set of Lévy densities ${\omega }_0$ which are supported in $[0,\infty )$ and satisfy the log-condition; additionally, define the ladder operator $\mathcal{I}$ by (40): if $n\ge 0$, then ${\omega }_{n+1}=\mathcal{I}\left({\omega }_n\right)$. Define sets ${\mathcal{W}}_n$ of SD Lévy densities by

$${\mathcal{W}}_{n+1}=\{{\omega }_{n+1}=\mathcal{I}\left({\omega }_n\right):{\omega }_n\in {\mathcal{W}}_n,n\ge 1\}.$$

The next result gives an integral representation for ${\omega }_n$ and a very simple expression for r-order cumulants along the ladder.

Theorem 8. 

Suppose that

$${\int }_1^{\infty }{\left(\log z\right)}^n{\omega }_0\left(z\right)dz<\infty ,(n\ge 0).$$

(49)

Then, ${\omega }_n$ is specified by

$${\omega }_n\left(x\right)={\displaystyle \frac{1}{x\Gamma \left(n\right)}}{\int }_x^{\infty }{(\log z-\log x)}^{n-1}{\omega }_0\left(z\right)dz={\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_1^{\infty }{\left(\log v\right)}^{n-1}{\omega }_0\left(xv\right)dv$$

(50)

and is a Lévy density. In addition, if $r>0$ and the order-r cumulant of $L\left(X_0\right)$ is finite:

$${\kappa }_r={\int }_0^{\infty }{x}^r{\omega }_0\left(x\right)dx<\infty$$

(51)

then the order-r cumulant of $L\left(X_n\right)$ is

$${\kappa }_{n,r}:={\int }_0^{\infty }{x}^r{\omega }_n\left(x\right)dx=r^{-n}{\kappa }_r.$$

(52)

Proof. 

Clearly, (50) is valid if $n=1$. Assuming its truth for arbitrary $n\ge 1$, it follows from (40) that

$$\begin{array}{ccc}\hfill {\omega }_{n+1}\left(x\right)& =& {\displaystyle \frac{1}{x\Gamma \left(n\right)}}{\int }_x^{\infty }{y}^{-1}{\int }_y^{\infty }{(\log z-\log y)}^{n-1}{\omega }_0\left(z\right)dz\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_x^{\infty }{\omega }_0\left(z\right){\int }_x^z{(\log z-\log y)}^{n-1}{y}^{-1}dydz,\hfill \end{array}$$

where interchanging the repeated integrals is valid by Fubini’s theorem. The inner integral equals

$${\int }_x^z{(\log z-\log y)}^{n-1}{d}_y\log y=n^{-1}{(\log z-\log x)}^n.$$

Hence, by induction, (50) holds for all n. The hypothesis ensures that ${\omega }_n\left(x\right)<\infty$ for all n and $x>0$.

It follows by construction that ${\omega }_{n+1}$ is the Lévy density determined by the BDLP generated by ${\omega }_n$ provided that $I_n={\int }_1^{\infty }\left(\log x\right){\omega }_n\left(x\right)dx<\infty$. Evaluating as above,

$$I_n={\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_1^{\infty }{\omega }_0\left(z\right)\left({\int }_1^{\infty }{(\log z-\log x)}^{n-1}\log x{d}_x\log x\right)dz;$$

writing $\log x=\log z-(\log z-\log x)$, we see that

$$I_n={\displaystyle \frac{1}{(n+1)!}}{\int }_1^{\infty }{\omega }_0\left(z\right){\left(\log z\right)}^{n+1}dz<\infty .$$

Thus, if $n\ge 1$, then ${\omega }_n$ is the Lévy density of an SD law, as asserted.

Finally, the left-hand side of (52) can be expressed in a similar manner as ${(\Gamma \left(n\right))}^{-1}{\int }_0^{\infty }{\omega }_0\left(z\right)J\left(z\right)dz$, where

$$J\left(z\right)={\int }_z^{\infty }{x}^{r-1}{\left(\log (z/x)\right)}^{n-1}dx=\Gamma \left(n\right)z^r{r}^{-n}$$

and (52) follows. □

Remark 3. 

It is reported in [9] (the original source [29] is unavailable to the author) that if $L\left(X\right)\in L_n^{+}$, then $X\stackrel{L}{=}{\int }_0^{\infty }{e}^{-t}d{S}_0(t^n/n!)$. This identity gives a direct but longer proof of (50).

Note that if $r>1$ in (52), then the cumulants along the ladder satisfy ${\kappa }_{nr}\to 0$ as $n\to \infty$. This manifests the nesting property of the classes $L_n^{+}$.

We next describe some consequences of (52). Referring to (51), assume that ${\kappa }_1<\infty$; also note from (52) that ${\kappa }_{n1}={\kappa }_1$. We assume, with no loss of generality, that ${\kappa }_1=1$. It follows that

$${\overline{\omega }}_n\left(x\right)={\int }_x^{\infty }{\omega }_n\left(y\right)dy,(n\ge 0)$$

is a density function, say of $A_n$, with LST $C_n\left(\theta \right)/\theta$. Denote the LST of $X_n$ by $T_n\left(\theta \right)=\exp (-C_n\left(\theta \right))$. It follows from (41) that

$$-T_{n+1}^{\prime }\left(\theta \right)=C_{n+1}^{\prime }\left(\theta \right)T_{n+1}\left(\theta \right)={\theta }^{-1}{C}_n\left(\theta \right)T_{n+1}\left(\theta \right).$$

This identity has the random variable expression

$${\widehat{X}}_{n+1}\stackrel{L}{=}{X}_{n+1}+A_n,(n\ge 0),$$

(53)

where ${\widehat{X}}_n$ is the length-biased version of $X_n$, i.e., noting that $E\left(X_n\right)=1$, then $P({\widehat{X}}_n\in dx)=xP(X_n\in dx)$.

A relation (see [12] ) of the type (53) with arbitrary $L\left(A_n\right)$ characterizes the infinite divisibility of positive laws having a finite first moment; here, we see that it holds along the ladder.

Next, still assuming that ${\kappa }_1=1$, we may set $r=1+𝚤\varrho$ in (52), where $𝚤=\sqrt{-1}$, observing that ${(1+𝚤\varrho )}^{-1}$ is the characteristic function of the standard exponential law $L\left(ϵ\right)$ and that ${\kappa }_{n,1+𝚤\varrho }$ is the characteristic function of $\log W_n$, where, for $n\ge 0$, $W_n$ has the density function $k_n\left(x\right)=x{\omega }_n\left(x\right)$. Then, since $U=e^{-ϵ}$ has the standard uniform law, we obtain the following random variable representation of (52), which has a central rôle in proving the limit Theorems 10–12.

Theorem 9. 

If ${\kappa }_1=1$, then

$$W_n\stackrel{L}{=}{V}_n{W}_0(n\ge 0),$$

where $V_n=U_1\times \cdots \times U_n$ is the product of independent copies of U.

Example 7. 

Suppose that ${\gamma }_t\sim \mathrm{gamma}\left(t\right)$, i.e., its density function is ${(\Gamma \left(t\right))}^{-1}{x}^{t-1}{e}^{-x}$. A standard computation shows that the density function of the product $ϵe^{-{\gamma }_t}$ is

$$\begin{array}{ccc}\hfill g_t\left(x\right)& =& {\displaystyle \frac{1}{\Gamma \left(t\right)}}{\int }_1^{\infty }{e}^{-xy}{\left(\log y\right)}^{t-1}{y}^{-1}dy,(t>0)\hfill \\ & =& {\displaystyle \frac{x}{\Gamma (1+t)}}{\int }_1^{\infty }{e}^{-xy}{\left(\log y\right)}^{t}dy(t\ge 0).\hfill \end{array}$$

Thus, $g_0\left(x\right)=e^{-x}$, as expected, and $g_1\left(x\right)=E_1\left(x\right)$, the exponential integral.

In Examples 5 and 6 it can be seen that $W_0={\gamma }_2^{\prime }/Z$, where ${\gamma }_2^{\prime }\sim \mathrm{gamma}\left(2\right)$ and $Z>0$. Thus, $P(Z\in ds)={\delta }_1\left(ds\right)$ for the gamma ladder and $P(Z\in ds)=s^{-1}\nu \left(ds\right)$ for the Lambert ladder. Now, $V_n\stackrel{L}{=}{V}_{n-1}U\stackrel{L}{=}{e}^{-{\gamma }_{n-1}}U$ and $U{\gamma }_2^{\prime }\stackrel{L}{=}ϵ$, so $W_n\stackrel{L}{=}{e}^{-{\gamma }_{n-1}}ϵ/Z$; hence, the density function of $W_n$ is

$$x{\omega }_n\left(x\right)={\int }_0^{\infty }s{g}_{n-1}\left(xs\right)d{F}_Z\left(s\right).$$

It follows from the form of $g_t$ that $x{\omega }_n\left(x\right)$ is completely monotone; hence, all members $L\left(X_n\right)$ of the ladder are generalized gamma convolutions.

We now show how Theorem 9 provides information about the nature of $L\left(X_n\right)$ when n is large. Theorem 10 shows that $L\left(X_n\right)$ is asymptotically degenerate provided that ${\kappa }_1<\infty$. Theorem 11 is a central limit convergence rate result which holds if ${\kappa }_r<\infty$ for some $r>2$. Theorem 12 concerns cases where ${\kappa }_1=\infty$. The basic assumption is that the CF $C_0\left(\theta \right)$ is regularly varying at zero, and the outcome is that there is a sequence of positive constants $b_n\to 0$ such that $\left(b_n{X}_n\right)$ has a stable law limit.

Theorem 10. 

Suppose that $a:={\kappa }_1<\infty$. Then, $X_n\stackrel{L}{\to }a$ as $n\to \infty$.

Proof. 

We can normalize $k_n\left(x\right)$ to be a density function; writing $W=W_0$ in Theorem 9, this has the result that the CF of $L\left(X_n\right)$ is

$$\begin{array}{ccc}\hfill C_n\left(\theta \right)& =& a{\int }_0^{\infty }\left(1-e^{-\theta x}\right)x^{-1}{k}_n\left(x\right)dx\hfill \\ & =& aE\left[{\displaystyle \frac{1-\exp (-\theta V_{n}W)}{V_{n}W}}\right].\hfill \end{array}$$

(54)

However, $V_n\to 0$ almost surely, and $x^{-1}(1-e^{-x})={\int }_0^1{e}^{-xy}dy\le 1$, so it follows from the dominated convergence theorem that $C_n\left(\theta \right)\to a\theta$, the CF of the unit mass at a. □

The next result shows under a mild additional moment condition that fluctuations of $L\left(X_n\right)$ around a are normally distributed.

Theorem 11. 

Suppose that ${\kappa }_r<\infty$ for some $r>2$. Then, with $a={\kappa }_1$ and ${\sigma }^2={\kappa }_2$,

$${\mathcal{N}}_n:=2^{n/2}(X_n-a)\stackrel{L}{\to }\sqrt{a}\sigma \mathcal{N},$$

(55)

where $L\left(\mathcal{N}\right)$ is the standard normal law.

Proof. 

The negative of the cumulant function of ${\mathcal{N}}_n$ is $a{2}^{n/2}{\beta }_n\left(\theta \right)$, where

$${\beta }_n\left(\theta \right)=\theta -E\left[{\displaystyle \frac{1-\exp (-\theta Q_n)}{Q_n}}\right]$$

and $Q_n=2^{n/2}{V}_{n}W$. Thus,

$${\beta }_n\left(\theta \right)={\int }_0^{\theta }{\xi }_n\left(y\right)dy\mathrm{where}{\xi }_n\left(\theta \right)=E\left[1-e^{-\theta Q_n}\right].$$

Since $E\left(Q_n\right)=2^{-n/2}E\left(W\right)=2^{-n/2}{\sigma }^2$, we have

$${\xi }_n\left(\theta \right)=\theta 2^{-n/2}{\sigma }^2-R_n\left(\theta \right),$$

where

$$\begin{array}{ccc}\hfill 0\le R_n\left(\theta \right)& =& E\left[\theta Q_n-\left(1-e^{-\theta Q_n}\right)\right]\hfill \\ & =& \theta E\left[{\int }_0^{Q_n}\left(1-e^{-\theta y}\right)dy\right]\hfill \\ & \le & \theta E\left[Q_n\left(1-e^{-\theta Q_n}\right)\right].\hfill \end{array}$$

We denote the last expectation by ${\overline{R}}_n\left(\theta \right)$ and estimate it as follows.

Note first that if $\epsilon <1$, then $1-e^{-x}\le \epsilon$ if $x\le \delta :=-\log (1-\epsilon )$. Hence,

$$E\left[Q_n\left(1-e^{-\theta Q_n}\right);Q_n\le \delta /\theta \right]\le \epsilon E\left(Q_n\right)=\epsilon 2^{-n/2}{\sigma }^2.$$

Next, set ${\delta }^{\prime }=\delta /\theta$ and observe that

$$E(Q_n;Q_n>{\delta }^{\prime })={\delta }^{\prime }{\overline{F}}_n\left({\delta }^{\prime }\right)+{\int }_{{\delta }^{\prime }}^{\infty }{\overline{F}}_n\left(x\right)dx,$$

(56)

where ${\overline{F}}_n\left(x\right)=P(Q_n>x)$. Markov’s inequality yields ${\overline{F}}_n\left(x\right)\le x^{-1}E\left(Q_n\right)\to 0$ as $n\to 0$. We need a much finer estimate of the rate of this convergence.

Note that with $c=\frac{1}{2}\log 2=0.3466$,

$${\overline{F}}_n\left(x\right)=P(V_n>x{2}^{-n/2}{W}^{-1})=P({\gamma }_n\le cn-\log x+\log W).$$

If $s>0$, then

$$E\left(e^{-s{\gamma }_n}\right)={(1+s)}^{-n}\ge E\left(e^{-s{\gamma }_n};{\gamma }_n\le z\right)\ge e^{-sz}P({\gamma }_n\le z),$$

i.e., $P({\gamma }_n\le z)\le e^{sz}{(1+s)}^{-n}$, a Chernoff type inequality. Taking $z=cn-\log x+\log W$, we obtain

$${\overline{F}}_n\left(x\right)\le {(1+s)}^{-n}{e}^{scn}{x}^{-s}E\left(W^s\right).$$

(57)

Choosing $s>1$, we find that the integral term $I_n\left(\theta \right)$ in (56) satisfies

$$I_n\left(\theta \right)\le {(s-1)}^{-1}{(\theta /\delta )}^{s-1}{(1+s)}^{-n}{e}^{scn}E\left(W^s\right).$$

The n-dependent factor is $\exp (-nb(s\left)\right)$, where $b\left(s\right)=\log (1+s)-cs$. Clearly, $b\left(1\right)=\log 2-c=c>0$, $b^{\prime }\left(1\right)=\frac{1}{2}-c>0$, and $b^{\prime \prime }\left(s\right)=-{(1+s)}^{-2}<0$; hence, b is concave-increasing in a neighbourhood of $s=1$.

Observing that our cumulant function equals $a{e}^{cn}{\beta }_n\left(\theta \right)$, we must consider the quantity

$$\exp (cn-nb(s\left)\right)=\exp [-n(\log (1+s)-cs-c\left)\right].$$

However, $b\left(1\right)-c=\log 2-2c=0$ and $b^{\prime }\left(1\right)>0$, so choosing $s\in (1,r-1)$, we have ${\int }_{{\delta }^{\prime }}^{\infty }{x}^{-s}dx<\infty$ and $E\left(W^s\right)<\infty$; hence, we have the estimate $I_n\left(\theta \right)={\theta }^{s-1}{\eta }_n$, where ${\eta }_n\to 0$ is independent of $\theta$, and it follows that ${\int }_0^{\theta }{I}_n\left(y\right)dy\to 0$. The bound (57) implies that ${\delta }^{\prime }{\overline{F}}_n\left({\delta }^{\prime }\right)=O\left({\theta }^{s-1}\right){\eta }_n^{\prime }$, where ${\eta }_n^{\prime }\to 0$ is independent of $\theta$.

It follows from these estimates that ${\int }_0^{\theta }E(Q_n;Q_n>\delta /y)dy\to 0$, and hence that

$$0\le \underset{n\to \infty }{\lim \sup } 2^{n/2}{\int }_0^{\theta }{R}_n\left(y\right)dy\le \frac{1}{2}\epsilon {\sigma }^2{\theta }^2.$$

Combining these estimates yields

$$\frac{1}{2}a{\sigma }^2{\theta }^2\le a \underset{n\to \infty }{\lim \sup } 2^{n/2}{\beta }_n\left(\theta \right)\le \frac{1}{2}(1+\epsilon )a{\sigma }^2{\theta }^2.$$

Since $\epsilon$ is arbitrary, we conclude that ${\lim }_{n\to \infty }E\left[\exp (-\theta {\mathcal{N}}_n)\right]=\exp \left(\frac{1}{2}a{\sigma }^2{\theta }^2\right)$, and the assertion follows from a continuity theorem for moment-generating functions, e.g., Theorem 1 in [30] or p. 408 in [25]. □

Let $0<\alpha \le 1$ and suppose that $C_0$ is regularly varying at zero, i.e.,

$$C_0\left(\theta \right)={\theta }^{\alpha }\lambda \left({\theta }^{-1}\right),$$

(58)

where $\lambda \left(x\right)$ is slowly varying (at infinity). This means that for any constant $c\in (0,\infty )$, we have ${\lim }_{x\to \infty }(\lambda \left(cx\right)/\lambda \left(x\right))=1$. A representation theorem asserts that there is a positive constant A, a measurable index function $\delta \left(x\right)\to 0$ as $x\to \infty$, and

$$\lambda \left(x\right)\sim A\exp \left({\int }_1^x\delta \left(y\right)y^{-1}dy\right).$$

(59)

See Bingham et al. [18] for these concepts.

We will use a representation motivated by expressing (50) as

$${\omega }_n\left(x\right)=E\left[Y_n^2{\omega }_0\left(Y_{n}x\right)\right],$$

where $Y_n\ge 1$ has the density function ${\left(\log v\right)}^{n-1}/(v^2\Gamma \left(n\right))$ supported by $[1,\infty )$. In fact, $Y_n=\exp \left({\gamma }_n\right)$, where we will assume that the gamma random variable ${\gamma }_n$ is the sum of n independent copies of $ϵ\sim Exp\left(1\right)$, ensuring that $Y_n\uparrow \infty$ almost surely. Integration yields the representation

$$C_n\left(\theta \right)=E\left[Y_n{C}_0(\theta /Y_n)\right],$$

(60)

which holds for any positive infdiv $L\left(X_0\right)$ satisfying the log-condition.

If ${\overline{\rho }}_0\left(x\right)$ is the right-hand tail of the Lévy measure of $L\left(X_0\right)$ and if $0<\alpha <1$, then (58) is equivalent to the following: as $x\to \infty$,

$${\overline{\rho }}_0\left(x\right)\sim {\displaystyle \frac{x^{-\alpha }\lambda \left(x\right)}{\Gamma (1-\alpha )}}.$$

We can now state the following result about convergence to a stable law. Note the quite specific form assumed for the slowly varying factor in (61).

Theorem 12. 

Assume that (58) holds. (i) Suppose that $0<\alpha <1$ and that there is a real number β such that, for $x>1$,

$$\lambda \left(x\right)={\left(\log x\right)}^{\beta }\ell \left(\log x\right),$$

(61)

where ℓ is slowly varying. Then, there exist norming constants

$$b_n=C_0^{-1}\left({\alpha }^n\right)\sim {\alpha }^{n/\alpha }\tilde{\lambda }\left({\alpha }^{-n}\right),$$

(62)

where $\tilde{\lambda }$ is slowly varying, and

$$L\left(b_n{X}_n\right)\stackrel{L}{\to }\mathrm{stable}(\alpha ,K_{\beta }),$$

(63)

where

$$K_{\beta }={(1-\log \alpha )}^{-1}{)}^{\beta }.$$

(ii) If (58) holds with $\alpha =1$ and (61) holds with $\beta \ge 0$, then there exist norming constants $b_n\sim n^{-\beta }\tilde{\lambda }\left(n^{\beta }\right)$, where $\tilde{\lambda }$ is slowly varying, such that $b_n{X}_n\stackrel{L}{\to }1$.

Proof. 

(i) Supposing that $0<b_n\to 0$, it follows from (58), (60), and (61) that the CF of $b_n{X}_n$ is

$$\begin{array}{ccc}\hfill C_n\left(b_n\theta \right)& =& {\left(\theta b_n\right)}^{\alpha }E\left[Y_n^{1-\alpha }\lambda (Y_n/\theta b_n)\right]\sim {\left(\theta b_n\right)}^{\alpha }E\left[Y_n\lambda (Y_n/b_n)\right]\hfill \\ & \sim & {\theta }^{\alpha }{C}_0\left(b_n\right)E[Y_n^{1-\alpha }\lambda (Y_n/b_n)/\lambda (1/b_n)].\hfill \end{array}$$

Denoting the expectation factor by $J_n$ and writing ${\nu }_n=\log \left(b_n^{-1}\right)$, it follows from (61) that the slowly varying quotient can be replaced by

$$Q_n={\left(1+{\displaystyle \frac{{\gamma }_n}{{\nu }_n}}\right)}^{\beta }{\displaystyle \frac{\ell ({\gamma }_n+{\nu }_n)}{\ell \left({\nu }_n\right)}}.$$

Choosing $b_n$ as in the first equality of (62), the asymptotic form there follows from results about the asymptotic inverses of regularly varying functions (p. 28 in [18]). In addition, since $\left(\log \tilde{\lambda }\left(x\right)\right)/\left(\log x\right)\to 0$ (p. 16 in [18]), it follows from (62) that

$$\underset{n\to \infty }{\lim }{\nu }_n/n=\rho >0.$$

(64)

Consequently, the slowly varying quotient in $Q_n$ converges almost surely to unity by virtue of the strong law and the uniform convergence property of slowly varying functions. Note also that given $\epsilon >0$, Potter’s bounds imply that this quotient is bounded above by a factor proportional to ${(1+{\gamma }_n/{\nu }_n)}^{\epsilon }$ provided that n is sufficiently large.

Next, define the parameter $\rho$ by $\alpha \rho =\log \alpha$ and observe that

$$\begin{array}{ccc}\hfill E\left[Y_n^{1-\alpha }{\left(1+{\displaystyle \frac{{\gamma }_n}{\rho n}}\right)}^{\beta }\right]& =& {\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_0^{\infty }{x}^{n-1}{\left(1+{\displaystyle \frac{x}{\rho n}}\right)}^{\beta }{e}^{-\alpha x}dx\hfill \\ & =& {\displaystyle \frac{{\alpha }^{-n}}{\Gamma \left(n\right)}}{\int }_0^{\infty }{y}^{n-1}{\left(1+{\displaystyle \frac{y}{\alpha \rho n}}\right)}^{\beta }{e}^{-y}dy\hfill \\ & =& {\alpha }^{-n}E{\left(1+{\displaystyle \frac{{\gamma }_n}{\alpha \rho n}}\right)}^{\beta }\sim {\alpha }^{-n}{K}_{\beta },\hfill \end{array}$$

appealing to the strong law and uniform integrability. Combining these estimates, the dominated convergence theorem leads to the conclusion that $J_n\sim K_{\beta }{\alpha }^{-n}$. From (62), we have $C_0\left(b_n\right)={\alpha }^n$, so it follows that $C_n\left(b_n\theta \right)\to K_{\beta }{\theta }^{\alpha }$, whence (63).

For (ii), observe first that $\beta \ge 0$ if $\alpha =1$. Changing some details in the above argument, we obtain $C_n\left(\theta \right)\sim \theta C_0\left(b_n\right)J_n$, where we now have $J_n\sim E\left[{(1+{\gamma }_n/{\nu }_n)}^{\beta }\right]$. Anticipating that ${\nu }_n=o\left(n\right)$, it follows from the mean value theorem that $J_n\sim E\left[{({\gamma }_n/{\nu }_n)}^{\beta }\right]\sim {(n/{\nu }_n)}^{\beta }$. However, $C_0\left(b_n\right)=b_n{\nu }_n^{\beta }\tilde{\ell }\left(b_n^{-1}\right)$, where $\tilde{\ell }\left(x\right)=\ell \left(\log x\right)$, a slowly varying function. Hence, $C_n\left(b_n\theta \right)\sim \theta b_n\tilde{\ell }\left(b_n^{-1}\right)n^{\beta }\to \theta$ if $b_n$ is chosen as in the assertion. In addition, ${\nu }_n\sim \beta \log n$, so the above provisional assumption does hold and the assertion follows. □

In Theorem 12, the case with $\alpha =1$ and $\beta =0$ generalises Theorem 10. If $0<\alpha <1$ and $\ell \left(x\right)\to a>0$ in (61), then the index function in (59) has the form $\delta \left(x\right)=(\beta /\log x)+r\left(x\right)$, where ${\int }_1^{\infty }{x}^{-1}r\left(x\right)dx$ is finite. Computation shows that $b_n\sim M{\left({\alpha }^n{n}^{-\beta }\right)}^{1/\alpha }$, where $M={\left(a{\rho }^{\beta }\right)}^{-1/\alpha }$. Thus, the case $\beta =0$ here generalises the final assertion of Example 4.

Still assuming that $0<\alpha <1$ and (59), we can obtain some idea of what can occur when $\left(\log x\right)\delta \left(x\right)\to \pm \infty$. Omitting details, it is clear that

$$J_n\sim E\left[Y_n^{1-\alpha }{e}^{{\nu }_n{I}_n({\gamma }_n/{\nu }_n)}\right],$$

where $I_n\left(x\right)={\int }_1^{1+x}\delta \left(b_n^{-z}\right)dz$ and $x>0$. Then, the change of gamma scaling used in the proof of Theorem 12 now yields

$$J_n\sim {\alpha }^{-n}E[\exp \left({\nu }_n{I}_n({\gamma }_n/\alpha {\nu }_n)\right].$$

Anticipating that ${\nu }_n\sim \rho n$, the strong law implies that $I_n({\gamma }_n/\alpha \nu )\sim I_n(1/\alpha \rho )=o\left(1\right)$ as $n\to \infty$.

Suppose further that $\delta \left(b_n^{-z}\right)\sim {\sigma }_n\psi \left(z\right)$, where ${\sigma }_n\to 0$, and that $\eta ={\int }_1^{1+1/\alpha \rho }\psi \left(z\right)dz$ is finite. It follows that the essential outcome of Theorem 12(i) occurs if $b_n$ is chosen to satisfy $C_0\left(b_n\right){\alpha }^{-n}\exp \left(\eta {\nu }_n{\sigma }_n\right)\to 1$. Two examples illustrate what can occur.

Example 8. 

If $0<r<1$ and $\delta \left(x\right)=A{\left(\log x\right)}^{r-1}$, then $\psi \left(z\right)=A{z}^{r-1}$ and ${\sigma }_n={\nu }_n^{r-1}$. Noting that $\lambda \left(x\right)\sim a\exp \left((A/r){\left(\log x\right)}^r\right)$, it turns out that the norming sequence can be taken as

$$b_n\sim {\left[a^{-1}{\alpha }^n\exp (-M{n}^r)\right]}^{1/\alpha },$$

where $M=(A/r){\rho }^r{K}_r$. The slower rate of decrease of the index function is reflected in a more quickly decaying norming sequence.

Example 9. 

If $\delta \left(x\right)=A\left(\log \log x\right)/\log x$, where $x>e$, then $\psi \left(z\right)=A/z$ and ${\sigma }_n=\left(\log {\nu }_n\right)/{\nu }_n$. In addition, $\lambda \left(x\right)\sim a\exp \left[\frac{1}{2}A{\left(\log \log x\right)}^2\right]$. It follows that

$$b_n=const.{\left[{\alpha }^n{n}^{-A\log (\alpha +1/\rho )}\exp (-\frac{1}{2}A{\left(\log n\right)}^2)\right]}^{1/\alpha }.$$

We next obtain some representations motivated by the resemblance of the expression for ${\omega }_n\left(x\right)$ in (50) to some formulae derived by Urbanik [10]. He begins with the class $L_0$ of SD laws on the real line. His class $L_1$ comprises all limit laws of weakly convergent normed sums $m^{-1}{\sum }_{j=1}^m{\xi }_j-a_m$, where the summands are independent, $L\left({\xi }_j\right)\in L_0$ for $j\ge 1$, and $\left\{a_m\right\}$ is a sequence of centering constants. The class $L_2$ is constructed from $L_1$ in the same way, and so on, inducing classes of SD laws $L_{n+1}\subset L_n$, where the set containment is proper. An important characterisation result (Proposition 1 in [10]) asserts that $L\left(X\right)\in L_{n+1}$ if and only if there exists $L\left(X_c\right)\in L_n$ for each $c\in (0,1)$ such that $X\stackrel{L}{=}cX+X_c$. This result implies a stochastic integral representation of the form (39), where the integrator is a Lévy process with one-dimensional laws in $L_n$.

Proposition 3 in [10] is a representation for the so-called spectral measure of the spectrally positive members of $L_n$. The proof uses a representation ((2) in [10]) for SD characteristic functions (derived earlier in [29]) and some theory of extreme points of convex sets. A parallel result for positive SD laws (with no drift term) is obtained by elementary means as follows. In keeping with our ladder context, we let $\omega \left(x\right)$ be the Lévy density of a positive SD law $L\left(X\right)$, so $k\left(x\right)=x\omega \left(x\right)$ is non-increasing and ${\int }_0^{\infty }(1\wedge x^{-1})k\left(x\right)dx<\infty$. Note that

$$\mathcal{E}\left(\theta \right)={\int }_0^{\theta }\left(1-e^{-y}\right)y^{-1}dy={\int }_0^1\left(1-e^{-\theta x}\right)x^{-1}dx$$

is the CF corresponding to the Lévy density $x^{-1}{1}_{(0<x\le 1)}$.

Lemma 5. 

The CF of a positive SD law has the representation

$$C\left(\theta \right)={\int }_0^{\infty }\mathcal{E}\left(\theta x\right){\displaystyle \frac{Q\left(dx\right)}{\log (1+x)}},$$

(65)

where Q is the finite measure on $(0,\infty )$,

$$Q\left(dx\right)=-\left[\log \right(1+x\left)\right]dk\left(x\right).$$

Proof. 

We have

$$C\left(\theta \right)={\int }_0^{\infty }\left(1-e^{-\theta x}\right)x^{-1}k\left(x\right)dx={\int }_0^{\infty }k\left(x\right)d_x\mathcal{E}\left(\theta x\right).$$

Noting that $\mathcal{E}\left(\theta \right)\sim \log \theta$ as $\theta \to \infty$, integration by parts yields

$$C\left(\theta \right)=-{\int }_0^{\infty }\mathcal{E}\left(\theta x\right)dk\left(x\right),$$

(66)

provided that ${\lim }_{x\to \infty }\left(\log x\right)k\left(x\right)=0$. However, the standing condition ${\int }_1^{\infty }w\left(x\right)dx<\infty$ implies that $I\left(x\right):={\int }_x^{\infty }{y}^{-1}k\left(y\right)dy\to 0$. If $\delta \in (0,1)$ and $x>1$, then $I\left(x\right)\ge {\int }_{x^{\delta }}^x{y}^{-1}k\left(y\right)dy\ge k\left(x\right)(1-\delta )\log x$; hence, (66) indeed holds, i.e., (65) holds.

To see that Q is finite, note that $k(\infty )=0$; otherwise, ${\int }_1^{\infty }{y}^{-1}k\left(y\right)dy=\infty$, so

$$Q(0,\infty )={\int }_0^{\infty }{\int }_0^x{\displaystyle \frac{dy}{1+y}}(-dk\left(y\right)={\int }_0^{\infty }{\displaystyle \frac{k\left(y\right)}{1+y}}dy<\infty .$$

□

Following Urbanik [10], Q is called the spectral measure of $L\left(X\right)$. He also defines an associated function, which here has the form

$$G_X\left(z\right)={\int }_{e^z}^{\infty }{\displaystyle \frac{Q\left(dx\right)}{\log (1+x)}}=k\left(e^z\right),(-\infty <z<\infty )$$

but which seems to have no useful role in our context.

Lemma 5 yields the following Urbanik-type representation of $C_n\left(\theta \right)$, which may be compared with Proposition 4 in [10].

Theorem 13. 

If ${\omega }_0$ satisfies (49), then for $n\ge 1$,

$$C_{n+1}\left(\theta \right)={\int }_0^{\infty }\mathcal{E}\left(\theta x\right){\omega }_n\left(x\right)dx,$$

and if $n\ge 1$, then

$${\omega }_n\left(x\right)={\displaystyle \frac{1}{x\Gamma \left(n\right)}}{\int }_x^{\infty }{\left(\log y-\log x\right)}^{n-1}{\displaystyle \frac{Q_1\left(dy\right)}{\log (1+y)}}.$$

Proof. 

The spectral measure for $L\left(X_n\right)$ is

$${\displaystyle \frac{Q_n\left(dx\right)}{\log (1+x)}}=-d_x{k}_n\left(x\right)=-{\displaystyle \frac{d}{dx}}\left(x{\omega }_n\left(x\right)\right)dx={\omega }_{n-1}\left(x\right)dx,$$

using (40). The first assertion is immediate, and the second is just a restatement of (50). □

The members of a BDLP ladder are related as in [10] by a quite specific limit relation. For easier notation, we render (39) as

$$X_{n+1}\stackrel{L}{=}{\int }_0^{\infty }{e}^{-t}d{S}_n\left(t\right).$$

(67)

Theorem 14. 

Assume that (49) holds. For each $n\ge 1$, $L\left(X_{n+1}\right)$ is the in-law limit (as $m\to \infty$) of the normed sum $X_{n+1,m}=m^{-1}{\sum }_{j=1}^m{\xi }_j\left(n\right)$, where ${\xi }_j\left(n\right)=j(S_n\left(H_j\right)-S_n\left(H_{j-1}\right))$ and $H_j$ is the j-th harmonic number.

Proof. 

The relation (41) can be written as

$$\begin{array}{ccc}\hfill C_{n+1}\left(\theta \right)& =& {\int }_0^1{z}^{-1}{C}_n\left(\theta z\right)dz=\underset{m\to \infty }{\lim }{\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m}{\left({\displaystyle \frac{j}{m}}\right)}^{-1}{C}_n\left(\theta {\displaystyle \frac{j}{m}}\right)\hfill \\ & =& \underset{m\to \infty }{\lim }\left[-\log E\left(\exp \left({\displaystyle \frac{-\theta }{m}}{\displaystyle \sum _{j=1}^m}j{S}_{n,j}\left(j^{-1}\right)\right)\right)\right],\hfill \end{array}$$

where $\{S_{n,j}\left(t\right);j\ge 1\}$ are independent copies of $S_n\left(t\right)$. The assertion follows from the continuity theorem for Laplace–Stieltjes transforms, the identity $S_n\left(j^{-1}\right)\stackrel{L}{=}{S}_n\left(H_j\right)-S_n\left(H_{j-1}\right)$, and the independence of the ${\xi }_j\left(n\right)$ for $j=1,2,\dots$ □

Thus, $L\left(X_{n+1}\right)$ is a weak limit in the Urbanik sense, with the precise identification (in our simpler setting) that the laws specifying the normed sums are scaled versions of one-dimensional laws of the BDLP, which is determined by $L\left(X_n\right)$.

We conclude by observing that (67) can be expressed as $X_{n+1}\stackrel{L}{=}{\int }_0^{1}y{d}_y{S}_n\left(\log y^{-1}\right)$ and that the integral is the limit of approximating sums

$${\tilde{X}}_{n+1,m}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=2}^m}j\left[S_n\left(-\log {\displaystyle \frac{j-1}{m}}\right)-S_n\left(-\log {\displaystyle \frac{j}{m}}\right)\right],$$

i.e., $L\left(X_{n+1}\right)$ is the limit of a triangular array having the required form.

7. A Branching Process Connection

A continuous-time and -space branching process (CB-process) $(Z_t:t\ge 0)$ is a Markov process with state space $[0,\infty )$ which satisfies the branching property: if $z>0$, then

$$E\left[\left.e^{-\theta Z_t}\right|Z_0=z\right]={\left\{E\left[\left.e^{-\theta Z_t}\right|Z_0=1\right]\right\}}^z.$$

It follows for each $t>0$ that $L\left(Z_t\right)$ is infdiv. CB-processes are constructed using a certain random-time transformation (the Lamperti transform, p. 261 in [31]) of a spectrally-positive Lévy process $(L_t:t\ge 0)$ where $L_0=Z_0=z>0$. We refer to this as the generating Lévy process (or subordinator), and denote its CF by $C\left(\theta \right)$. If the so-called Malthusian parameter $\mu =C^{\prime }\left(0\right)<\infty$, then $E\left(Z_t\right|Z_0=z)=z{e}^{\mu t}$.

The normed process $(M_t:t\ge 0)$, where $M_t=e^{-\mu t}{Z}_t$, is a martingale with respect to the natural filtration corresponding to the CB-process. We will assume that $\mu >0$, the CB-process is supercritical, and $C^{\prime \prime }\left(0\right)>-\infty$. In this case, $\left(M_t\right)$ converges almost surely and in mean-square to a random variable M which has an infinitely divisible law and $E\left(M\right)=Z_0$. The CF of $L\left(M\right)$, denoted by $\omega \left(\theta \right)$, solves the differential equation

$$\mu \theta {\omega }^{\prime }\left(\theta \right)=C\left(\omega \left(\theta \right)\right).$$

(68)

As noted in [17], the Lambert law is the CF of $L\left(M\right)$ in the case where $\left(L_t\right)$ is the compound Poisson law with unit rate and a unit exponential jump law. Here, $C\left(\theta \right)=\theta /(1+\theta )$, $\mu =1$, and the assertion follows from the above differential equation and (9). In particular, $C\left(\theta \right)$ is a particular case of the form (1) where the drift constant $\delta =0$.

Now, let $0<a\ne 1$ and recall that the infdiv law $L\left(X_a\right)$ of Section 4 has the LST $\chi (\theta ;a)=e^{-\omega (\theta ;a)}$. Again, suppressing explicit dependency on a, we have the identification ${\omega }^{\prime }\left(\theta \right)=-{\chi }^{\prime }\left(\theta \right)/\chi \left(\theta \right)$, and it follows from (35) and (22) that

$$\theta {\omega }^{\prime }\left(\theta \right)=a{\displaystyle \frac{{\chi }^{-1/a}\left(\theta \right)-{\chi }^{-1}\left(\theta \right)}{{\chi }^{-1/a}\left(\theta \right)-a{\chi }^{-1}\left(\theta \right)}};$$

hence, the differential equation in (68) is satisfied with

$$C\left(\theta \right)=a{\displaystyle \frac{e^{\theta /a}-e^{\theta }}{e^{\theta /a}-a{e}^{\theta }}}.$$

If this is a CF having the form in (1), then $L\left(X_a\right)$ is the law of a martingale limit law. Observe that $C^{\prime }\left(0\right)={\lim }_{\theta \downarrow 0}{\theta }^{-1}C\left(\theta \right)=1$, a consistent outcome.

Recalling the definition in (24) and defining $p=a^{-1}$ if $a>1$, it follows that

$$C\left(\theta \right)=\left\{\begin{array}{cc}a{\displaystyle \frac{1-e^{-\theta /q}}{1-a{e}^{-\theta /q}}}=a(1-g_a\left(e^{-\theta /q}\right)\hfill & \mathrm{if}a<1,\hfill \\ {\displaystyle \frac{1-e^{-\theta /q}}{1-p{e}^{-\theta /q}}}=1-g_p\left(e^{-\theta /q}\right)\hfill & \mathrm{if}a>1,\hfill \end{array}\right.$$

where, if $0<\alpha <1$, then $g_{\alpha }\left(s\right)=(1-\alpha )s/(1-\alpha s)$, which is the pgf of the geometric law with parameter $\alpha$ and ia supported on the natural numbers. Clearly, $C\left(\theta \right)$ is the CF of a compound Poisson process.

It follows that $L\left(X_a\right)$ is the martingale limit of a CB-process generated by a Lévy process with zero drift and the discrete Lévy measure

$$\mathcal{L}\left(dx\right)=\left\{\begin{array}{cc}a(1-a){\displaystyle {\sum }_{j=1}^{\infty }}{a}^{j-1}{\delta }_{j/q}\left(dx\right)\hfill & \mathrm{if}0<a<1,\hfill \\ (1-p){\displaystyle {\sum }_{j=1}^{\infty }}{p}^{j-1}{\delta }_{j/q}\left(dx\right)\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

This outcome further highlights the singularity of the Lambert law as a member of the family $\{L\left(X_a\right):a>0\}$; it is the only member associated with a CB-process generated by a Lévy process having a continuous Lévy measure. Observe that ${\lim }_{a\to 1}C(\theta ;a)=C(\theta ;1)=\theta /(1+\theta )$.

If $t>0$, then the CF of $\chi (\theta ;a,t)$ is $t\omega (\theta ;a)$. The latter satisfies (68) with $C(\theta ;a)$ replaced by $tC(\theta /t;a)$. Hence, if $0<a<1$, for example, then $\chi (\theta ;a,t)$ is the LST of the martingale limit of the CB-process generated by the compound Poisson process with Lévy measure

$${\mathcal{L}}_t\left(dx\right)=ta(1-a){\displaystyle \sum _{j=1}^{\infty }}{a}^{j-1}{\delta }_{j/qt}\left(dx\right).$$

If $qt=1$, then the Lévy measure is supported on the natural numbers. A similar outcome holds if $a>1$.

The lattice-like step function sample path structure of the above generating subordinators is preserved under the Lamperti transform. This prompts exploration of whether $\chi (\theta ;a,t)$ can be the LST of the martingale limit of a supercritical Markov branching process (MBP). This process (see Chapter 3 in [31]) models the evolution of the population size ${\mathcal{Z}}_t$ of a population of individuals whose lifetimes have an exponential law with parameter $\rho$ (the split rate) and $\zeta$ offspring at the end of these lives. The offspring law is the discrete law with masses $p_j=P(\zeta =j)$, $j=0,1,\dots$, where lifetimes and reproduction events are mutually independent. We assume that $0<p_0+p_1<1$ and that the mean number of offspring $m={\sum }_{j\ge 1}j{p}_j$ satisfies $1<m<\infty$. The Malthusian parameter for the MBP is $\mu =\rho (m-1)$ and $E\left({\mathcal{Z}}_t\right)={\mathcal{Z}}_0{e}^{\mu t}$. We assume that ${\mathcal{Z}}_0=1$. Finally, if ${\mathcal{Z}}_0=1$, then the probability of extinction $q_e$ is the least non-negative solution of $f\left(s\right)=s$, where $f\left(s\right)=E\left({\zeta }^s\right)$.

Here, if ${\sum }_{j\ge 1}{j}^2{p}_j<\infty$, then $M_t:=e^{-\mu t}{\mathcal{Z}}_t\to W$ almost surely and in mean-square, and the LST $\varphi \left(\theta \right)$ of W is determined by its functional inverse: if $q_e<s\le 1$, then

$${\varphi }^{-1}\left(s\right)=(1-s)\exp \left[-{\int }_s^1\left({\displaystyle \frac{m-1}{f\left(u\right)-u}}-{\displaystyle \frac{1}{1-u}}\right)du\right].$$

Taking $\varphi \left(\theta \right)=\chi (\theta ;a,t)$, it follows from (7) that ${\varphi }^{-1}\left(s\right)=\tau \left(s^{1/t}\right)$. Differentiation of the resulting identity yields

$$f\left(s\right)=s+(m-1)t{s}^{1-1/t}\tau \left(s^{1/t}\right)/{\tau }^{\prime }\left(s^{1/t}\right).$$

Logarithmic differentiation of (7) produces

$$\tau \left(s^{1/t}\right)/{\tau }^{\prime }\left(s^{1/t}\right)=\left\{\begin{array}{cc}-as{\displaystyle \frac{1-s^{1/q}}{1-a{s}^{1/q}}}\hfill & \mathrm{if}0<a<1,\hfill \\ -s{\displaystyle \frac{1-s^{1/q}}{1-p{s}^{1/q}}}\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

Observing that $f\left(0\right)=0$, it follows that if $f\left(s\right)$ is a probability generating function, then $q_e=0$. In addition, $f\left(s\right)$ can be the generating function of an offspring law only if t is chosen such that $qt={\overline{n}}^{-1}$, the reciprocal of a natural number $\overline{n}$. We assume this condition in what follows.

If $a<1$, we obtain the explicit algebraic form

$$f\left(s\right)=s-(m-1)ats·{\displaystyle \frac{1-s^{1/qt}}{1-a{s}^{1/qt}}}=p_{1}s+(1-p_1)(1-a){\displaystyle \sum _{j=1}^{\infty }}{a}^{j-1}{s}^{1+j\overline{n}},$$

(69)

where

$$p_1=1-(m-1)(1-a)/\overline{n}.$$

Thus, $p_1<1$; to ensure that $p_1\ge 0$, we choose the offspring mean m subject to

$$1<m\le 1+{\displaystyle \frac{\overline{n}}{1-a}}.$$

Assuming this condition, $f\left(s\right)$ is indeed the generating function of an offspring law with support $\mathcal{S}=\{1,1+\overline{n},1+2n,\dots \}$ and

$$P(\zeta =1+jn)=(1-p_1)(1-a)a^{j-1},(j=1,2,\dots ).$$

An admissible choice is $p_1=1-(m-1)/qn$, in which case $p_1=a$, i.e., the offspring law is geometric with parameter a and support $\mathcal{S}$.

If $a>1$, then $p_1=1-(m-1)/qn$; choosing $1<m\le 1+qn$, $f\left(s\right)$ has the form in (69), with p replacing a. In addition, a geometric law with parameter p supported on $\mathcal{S}$ results if $m=1+pqn=1+\overline{n}/(a-1)$, which is an admissible choice. Note that if $0<a\ne 1$, then $\chi (\theta ;a,1/qn)$ is the LST of the martingale limit of a family of supercritical MBPs in which each family is indexed by $m>1$, subject to the upper-bound restrictions described above.

Finally, if $a=1$, similar calculations lead to the identity

$${\displaystyle \frac{m-1}{f\left(s\right)-s}}=-{\left(st\right)}^{-1}(1+t/\log s).$$

(70)

The right-hand side is zero at $s=s^{*}=e^{-t}$, implying that $f\left(s\right)$ has a fixed point at $s^{*}$. Hence, if $f\left(s\right)$ is a probability generating function, then $f\left(0\right)>0$. However, the right-hand side of (70) tends to $-\infty$ as $s\downarrow 0$, which is a contradiction. Thus, the Lambert function case $a=1$ reveals its singularity by not corresponding to the martingale limit of a supercritical MBP.

8. Some Final Remarks

The results of Section 4 and Section 7 can be conceived in terms of an infinite fibre or thread of SD laws ${\mathcal{F}}_1=\{L\left(X_a\right):a>0\}$; in each of these, the elements are the martingale limit law of a supercritical CB-process. The element $L\left(X_1\right)$ of ${\mathcal{F}}_1$, for which the CF is $W\left(\theta \right)$, is the only element with a generating subordinator that has a continuous Lévy measure: its CF is $\theta /(1+\theta )$. Clearly, other such fibres can be constructed from any family of generating subordinators having a CF $C_a\left(\theta \right)$ indexed by a parameter $a>0$ such that $C_a\left(\theta \right)\to \theta /(1+\theta )$ as $a\to 1$. For example, $C_a\left(\theta \right)=1-{(1+\theta )}^{-a}$ corresponds to a compound Poisson process with gamma-distributed jumps. Some other examples can be found early in the dictionary of complete Bernstein functions in [28].

A detailed analysis of such a fibre, ${\mathcal{F}}_2=\{L\left(V_a\right):a\ge 1\}$, can be found in [17]. The formulation there is in terms of a parameter $q>0$, which here is given by $q=a-1$. The law $L\left(V_a\right)$ is the martingale limit law of the CB-process for which the generating subordinator has the CF

$$C_a\left(\theta \right)=\theta ·{\displaystyle \frac{1+(a-1)\theta }{1+a\theta }},$$

i.e., the generator is the compound Poisson process for which the drift parameter is $\delta =1-a^{-1}$ and the density of its Lévy measure is $a^{-3}{e}^{-x/a}$, i.e., exponentially distributed jumps.