On the Stationary Measure for Markov Branching Processes

Abstract

A previous study determined criteria ensuring that a probability distribution supported in positive integers is the limiting conditional law of a subcritical Markov branching process. It is known that there is an close connection between the limiting conditional law and the stationary measure of the transition semigroup. This paper revisits that theme of by seeking tractable criteria ensuring that a sequence on positive integers is the stationary measure of a subcritical or critical Markov branching process. These criteria are illustrated with several examples. The subcritical case motivates consideration of the Sibuya distribution, leading to the demonstration that members of a certain family of complete Bernstein functions, in fact, are Thorin–Bernstein. The critical case involves deriving a notion of the limiting law of population size given that extinction occurs at a precise future time. Examples are given, and some show an interesting relation between stationary measures and Hausdorff moment sequences.

1. Introduction

The Markov branching process $(Z_t:t\ge 0)$ is a Markov process with non-negative integer states which models the temporal progression of the size of a population of individuals whose independent lifetimes have an exponential law with the parameter (split rate) $a>0$. At the end of their lives, each individual begets j offspring with probability $p_j$, and all reproduction events are mutually independent and independent of life lengths. The probability-generating function (pgf) of this offspring number law is $f(s)={\sum }_{j=0}^{\infty }{p}_j{s}^j$, and we assume that $0<p_0<1$.

Sometimes, it is assumed that $p_1=0$ on the basis that begetting one offspring is unobservable, e.g., p. 118 in [1]. This assumption entails no loss of generality but, given our aims, we do not adopt it. We do assume that $p_0+p_1<1$. The transition probabilities $p_{ij}(t)=P_i(Z_t=j)$, where $P_i(·)$ and $E_i(·)$ signal that $Z_0=i$, are specified by the single pgf $F(s,t)=E_1(s^{Z_t})$ together with ${\sum }_{j=0}^{\infty }{p}_{ij}(t)s^j={(F(s,t))}^i$ expressing the independent evolution of lines of descent.

The zero state is accessible and absorbing, and hence, there is no limiting distribution. The transition semigroup does, however, have a stationary measure $({\pi }_j:j=1,2,\dots )$ in the sense that, for $t\ge 0$,

$$\sum _{i=1}^{\infty }{\pi }_i{p}_{ij}(t)={\pi }_j,(j\ge 1),$$

and this measure is unique up to multiplication by a constant; see p. 110 in [2]. We write $\pi \in SM$. As shall be explained in §2, we lose no generality by assuming that the mean per capita of offspring $m=f^{\prime }(1)={\sum }_{j=1}^{\infty }j{p}_j\le 1$; $(Z_t)$ is subcritical ($m<1$) or it is critical ($m=1$). In this case, the generating function (gf) of the stationary measure is

$$\pi (s)=\sum _{j=1}^{\infty }{\pi }_j{s}^j=K{\int }_0^s{\displaystyle \frac{du}{f(u)-u}},(0\le s<1)$$

(1)

where $K>0$ is an arbitrary constant. In particular, ${\pi }_1=K/p_0$. Recalling that $f(s)>s$ if $s\in [0,1)$, we see that $\pi (s)<\infty$.

In the case $m<1$, we choose $K=1-m$ and we write $f\in SC$ to mean f is a subcritical offspring-number pgf. The stationary measure is closely associated with discrete limiting conditional laws (LCLs) of the MBP. Let $T=inf\{t:Z_t=0\}$ denote the time to extinction, or hitting time of the zero state. Then, $P_i(T\le t)={(F(t))}^i$, where $F(t)=F(0,t)$ and $P_i(T<\infty )=1$, i.e., extinction is (almost) certain. If $m<1$, then the limits

$$b_j=\underset{t\to \infty }{lim}{P}_i(Z_t=j|T>t),(i,j\ge 1)$$

(2)

exist, they are independent of the initial population size i, ${\sum }_{j=1}^{\infty }{b}_j=1$, and their pgf is

$$B(s)=\sum _{j=1}^{\infty }{b}_j{s}^j=1-e^{-\pi (s)}.$$

(3)

The distribution (2) is called a limiting conditional law (LCL) and we write $B\in LCL$ to mean that $B(s)$ is the pgf of the LCL of some subcritical MBP, or that $(b_j)$ is an LCL, according to the context.

Until recently, very few explicit examples of LCLs were known. Theorem 2.1 in [3] (see Theorem 9 below) collects sufficient conditions (which are necessary) for a probability mass function $(b_j:j\ge 1)$ satisfying $b_1>0$ (written $B\in {\mathcal{B}}_{+}$), ensuring that $B\in LCL$. Several explicit examples have thereby been constructed. One motivation for this stems from the fact that $F(s,t)$ satisfies the identity

$$1-B(F(s,t))=(1-B(s))e^{-\mu t},$$

(4)

where $\mu =a(1-m)$. So, if $1-B(s)$ can explicitly be inverted, there results an explicit expression for $F(s,t)$. Several such examples are given in [3].

Not known to the authors at the time [3] was written is the fact that the earlier paper [4] presents a characterisation of LCLs in the form of a sequence of inequalities. This criterion seems less suited for constructing examples, though two are given. As an aside, the representations in [4] imply that $p_1=0$.

If $m=1$, the critical case, then we set $K=1$ in (1) and denote with $\mathcal{B}$ a Borel subset of the positive reals with finite Lebesgue measure $|\mathcal{B}|$. Theorem 7.1 in [5] asserts the existence of

$$\underset{t\to \infty }{lim}{P}_i(Z_t=j|T\in t+\mathcal{B})={\pi }_j{\sigma }_j,$$

(5)

where $({\sigma }_j)$ is a certain sequence such that ${\sigma }_j\to 0$ is geometrically fast as $j\to \infty$ and ${\sum }_{j=1}^{\infty }{\pi }_j{\sigma }_j=1$; see (50) below. Hence, the stationary measure looms large here too, and, in addition,

$$\pi (F(s,t))=\pi (s)+at.$$

(6)

Hence, explicit expressions for $\pi (s)$ and its inverse will yield explicit forms of $F(s,t)$.

The structure of this paper is as follows. In §2, we recall how the supercritical case can be reduced to the subcritical case. We then present characterisations of the stationary measure along the lines of Theorem 1 in [4] and of Theorem 2.1 in [3]. In §3, we begin by recalling a simple way of constructing laws concentrated on the positive integers which strictly contain the LCLs of subcritical MBPs; see Theorem 3. Thus, Theorem 3, in essence, is Theorem 1.1 in ([6]) but stated in a stronger form: If $({\pi }_j:j\ge 1)$ is a non-negative sequence satisfying (13) below, then $B(s)$ defined by (3) is an honest pgf if and only if (14) holds. In particular, if $j{\pi }_j\uparrow \rho \in (0,1]$, then $B(s;\rho )=1-{(1-B(s;1))}^{\rho }$ is a pgf. Theorem 4 gives estimates for the right-hand tails generated by $B(s;\rho )$.

The pgf $B(s;\rho )$ is the composition $h_{\rho }(B_1(s))$, where $h_{\rho }(s)=1-{(1-s)}^{\rho }$ is the pgf of the so-called Sibuya law [7]. This law is a shifted Poisson mixture whose infinitely divisible (infdiv) mixing law is known to be a generalised gamma convolution (see §3 for definitions) and, hence, is self-decomposable. This law also belongs to a larger family of infdiv laws whose Laplace exponents are complete-Bernstein functions. On the other hand, it has not been known (until now) whether they are Thorin–Bernstein. The remainder of §3 is a rather long detour (Theorems 5–8) which completely resolves this issue in the affirmative. We remark that in this portion of the text only, the letter a is used to denote one parameter which indexes the generalised Mittag–Leffler function.

The purpose of §4 is to supplement the conditions of Theorem 3 with additional structure, ensuring that $B(s)=B(s;1)\in LCL$. The discussion begins with general comments recalling the sense in which $B(s;\rho )$ generates a limiting conditional law if $B(s)\in LCL$. Then, Theorem 9 recalls Theorem 2.1 from ([3]) stated here in a stronger form. It gives workable criteria ensuring that $B\in LCL$. Examples 2 and 3 complete two examples in [3] by exhibiting explicit expressions for the stationary measure. Theorem 10 expresses Theorem 9 in terms of sequences $\pi \in {\mathsf{\Pi }}_{+}$. It is illustrated by an Example 5 which extends the treatment of the geometric offspring law in [8].

As motivation for §§5,6, let $(Y_n:n=0,1,\dots )$ denote a critical Galton–Watson process with time to extinction $\tilde{T}$. With no further moment conditions, it is shown in [9] (Theorem 1) that, for $k=1,2,\dots$,

$$\underset{n\to \infty }{lim}{P}_i(Y_n=j|\tilde{T}=n+k)={\lambda }_j(k)\mathrm{and}\sum _{j=1}^{\infty }{\lambda }_j(k)=1.$$

(7)

Theorem 2 in [9] asserts an evaluation of the limit which can be expressed as

$${\lambda }_j(k)={\phi }_j{P}_j(\tilde{T}=k),$$

(8)

where $({\phi }_j:j\ge 1)$ denotes the (unique up to scaling) stationary measure of the process; see p. 68 in [1]. In §5, we use (5) to obtain an MBP version (Theorem 11) of these results by attaching a meaning to ${lim}_{t\to \infty }{P}_i(Z_t=j|T=t+\tau )$, where $\tau >0$ is fixed. This limit turns out to belong to the power series laws generated by the weights $j{\pi }_j$.

Examples for Theorem 11 are presented in §6. If $f(s)$ is a critical offspring-law pgf, then $g(s)={\sum }_{j\ge 0}{g}_j{s}^j=(1-f(s))/(1-s)$ is a pgf and $g_j\downarrow 0$. The function $V(s)=1/(1-g(s))$ generates a renewal sequence $(v_j)$, a generalised one in the sense that $v_0>1$. In addition the stationary measure is determined by $p_{0}j{\pi }_j={\sum }_{i=0}^{j-1}{v}_i$. Theorem 12, a critical analogue of Theorems 9/10 is based on this relation. It draws from Example 6 which suggests that Hausdorff moment sequences may induce a law $(g_j)$ with the required monotonicity property. Examples 9 and 10 illustrate this idea. Alternatively, one may simply choose an admissible law $(g_j)$ to work back to a stationary measure. Example 11 exhibits this possibility.

2. Identification of Stationary Measures

Consider first a supercritical MBP, $1<m\le \infty$, where we choose the minimal construction if it has a finite explosion time; see p. 113 in [10]. Recalling that $p_0>0$, the probability of ultimate extinction q is the least positive solution of $f(s)=s$ and $q\in (0,1)$. Moreover, the offspring pgf satisfies $f(s)>s$ if $0\le s<q$ and $m_q:=f^{\prime }(q)<1$. The generating function $\pi (s)$ still has the form (1), but with $s\in [0,q)$; $\pi (q-)=\infty$. The gf $f_q(s)=q^{-1}f(qs)\in SC$. A change in variables in (1) yields

$${\pi }_q(s):=\pi (qs)=K{\int }_0^s{\displaystyle \frac{du}{f_q(u)-u}},$$

i.e., the sequence $({\pi }_j{q}^j)$ is the stationary measure of a subcritical MBP. Hence, it suffices to consider only the cases $m\le 1$.

Conversely, if $m<1$ and $f(s)$ has a radius of convergence so large that there exists $q<1$ such that $f(q^{-1})=q^{-1}$, then ${\widehat{f}}_q(s)=qf(s/q)$ is the offspring pgf of a supercritical MBP. Its stationary measure is ${\widehat{\pi }}_j=q^{-j}{\pi }_j$, where $\pi \in SM$ for the subcritical process. Hence, provided the above conditions hold, explicit examples for the subcritical case translate to the supercritical case.

Write $\pi \in {\mathsf{\Pi }}_{+}$ to mean that $({\pi }_j:j\ge 1)$ is a sequence of non-negative numbers with ${\pi }_1>0$, and that $\pi (s)={\sum }_{j=1}^{\infty }{\pi }_j{s}^j<\infty$ if $s\in [0,1)$ and ${\sum }_{j=1}^{\infty }j{\pi }_j=\infty$. The following result, similar in spirit to Theorem 1 in [4], gives conditions ensuring that $\pi \in SM$.

Theorem 1. 

(a) Suppose that $(p_j)$ is the offspring mass function of an MBP with offspring mean $m\le 1$ and an arbitrary splitting rate. Then, $p_0>0$, and the unique $\pi \in SM$, normalised so that $p_0{\pi }_1=1$ satisfies the recursive system

$$\sum _{i=0}^j{p}_{j-i}(i+1){\pi }_{i+1}={\delta }_{0j}+j{\pi }_j(j\ge 0)$$

(9)

In addition, $j{\pi }_j{\uparrow (1-m)}^{-1}\le \infty$ as $j\uparrow \infty$.

(b) Conversely, suppose that $(p_j:j\ge 0)$ is a non-negative sequence with $p_0>0$, and $p_1<1$ such that the system (9) has a solution $\pi \in {\mathsf{\Pi }}_{+}$. Then, $(p_j)$ is a mass function with mean $m\le 1$ and $\pi \in SM$. The corresponding MBPs are critical iff $j{\pi }_j\to \infty$.

Proof. 

If the conditions of (a) hold, then it is known that there is a unique stationary measure as asserted, e.g., p. 110 in [2]. In addition, setting $K=1$, its gf is $\pi (s)={\int }_0^s{(f(u)-u)}^{-1}du$, i.e.,

$$(f(s)-s){\pi }^{\prime }(s)=1.$$

(10)

Equating coefficients of $s^j$ in this identity yields (9). In addition, l’Hopital’s rule yields

$$\underset{s\to 1-}{lim}(1-s){\pi }^{\prime }(s)=\underset{s\to 1-}{lim}{\displaystyle \frac{1-s}{f(s)-s}}={\displaystyle \frac{1}{1-m}}.$$

Next, observe that

$$g(s)={\displaystyle \frac{1-f(s)}{m(1-s)}}=m^{-1}\sum _{j=0}^{\infty }{s}^j\sum _{i>j}{p}_i$$

(11)

is a pgf, and

$${\pi }^{\prime }(s)={\displaystyle \frac{1}{(1-s)(1-mg(s))}}$$

(12)

which implies that $(j{\pi }_j)$ is a non-decreasing sequence. It follows from the Tauberian theorem for power series ([11], p. 46) that $j{\pi }_j\to {(1-m)}^{-1}$. The whole of Assertion (a) follows.

For Assertion (b), let $a_j=j{\pi }_j$; multiply (9) by $s^j$, where $s\in (0,1)$; and sum over $j=0,\dots ,J<\infty$. This yields

$$\sum _{j=0}^J\sum _{i=0}^j{p}_{j-i}{a}_{i+1}{s}^j=\sum _{i=0}^J{a}_{i+1}{s}^i\sum _{j=i}^J{p}_{j-i}{s}^{j-i}=\sum _{i=0}^J{a}_{i+1}{s}^i\sum _{k=0}^{J-i}{p}_k{s}^k=1+\sum _{j=1}^J{a}_i{s}^j.$$

For a fixed number $J^{\prime }<J$, the left-hand side is bounded below by ${\sum }_{i=0}^{J^{\prime }}{a}_{i+1}{s}^i{\sum }_{k=0}^{J-i}{p}_k{s}^k$. Letting $J\to \infty$ on both sides of the resulting inequality and writing $f(s)={\sum }_{j=0}^{\infty }{p}_j{s}^j$ yields $f(s){\sum }_{i=0}^{J^{\prime }}{a}_i{s}^i\le 1+s{\pi }^{\prime }(s)<\infty$. Hence, $f(s)<\infty$.

It follows from the monotone convergence theorem that ${\pi }^{\prime }(s)f(s)=1+s{\pi }^{\prime }(s)$. This implies that $f(s)>s$ for $s\in [0,1)$ and that $f(1-)=1+1/{\pi }^{\prime }(1-)$. Hence, $f(s)$ is a pgf if and only if ${\sum }_{j}j{\pi }_j=\infty$, as assumed, and its mean $m=f^{\prime }(1)\le 1$. It follows that there is a family of non-supercritical MBPs with offspring pgf $f(s)$, and part (a) is applicable, implying that $\pi \in SM$ with ${\pi }_1{p}_0=1$. Finally, if $j{\pi }_j\to \infty$, then $(1-s){\pi }^{\prime }(s)\to \infty$ ($s\uparrow 1$), and hence, it follows from (10) that $(f(s)-s)/(1-s)\to 0$, i.e., $m=1$. □

Remark 1. 

That the sequence $(j{\pi }_j)$ is non-decreasing is proven in [12] using an elementary but longer argument based on (10).

A second characterisation is imbedded in the proof of Theorem 1 via (10).

Theorem 2. 

The sequence $({\pi }_j:j\ge 0)$ satisfying ${\pi }_0=0\le {\pi }_j$ for $j\ge 1$ is the stationary measure of a non-supercritical MBP if and only if the gf ${\pi }^{\prime }(s)={\sum }_{j=1}^{\infty }j{\pi }_j{s}^{j-1}$ has the form (12), where $0<m\le 1$, and $g(s)$ is a pgf whose masses $g_j$ comprise a non-decreasing sequence.

Proof. 

It suffices to observe that if m and $g(s)$ are as specified; then,

$$f(s)=1-m(1-s)g(s)=1-m{g}_0+m\sum _{j=1}^{\infty }\left(g_{j-1}-g_j\right)s^j$$

is a pgf. In addition, it follows from (12) that $f^{\prime }(1-)=m$. □

3. A General Criterion

In this section, we discuss a general criterion for constructing a class of laws on the positive integers which strictly contain the LCLs of subcritical MBPs. This material draws on [6], and the next result tightens Theorem 1.1 in that reference.

Theorem 3. 

If $\pi \in {\mathsf{\Pi }}_{+}$ satisfies

$$j{\pi }_j\le (j+1){\pi }_{j+1},(j\ge 1),$$

(13)

then $B(s)$ defined by (3) is a pgf if and only if

$$j{\pi }_j\le 1,(j\ge 1).$$

(14)

Proof. 

That these two conditions are sufficient is the content of Theorem 1.1 in [6]. For the converse, it is obviously necessary that $\pi (s)<\infty$ for $s\in (0,1)$. Assuming this condition and (13), it follows that if (14) does not hold, then we can choose $ϵ>0$ and then a natural number $j^{\prime }$ such that $j{\pi }_j>1+ϵ$ if $j>j^{\prime }$. Hence, $\pi (s)>\pi (s,j^{\prime })-(1+ϵ)log(1-s)$, where $\pi (s,j^{\prime })$ is a polynomial of degree $j^{\prime }$, implying that $1-B(s)<e^{-\pi (s,j^{\prime })}{(1-s)}^{1+ϵ}$, i.e., $B(s)$ is not a pgf. □

Remark 2. 

We assume in the sequel that (13) and (14) both hold and that ${\pi }_j>0$ for some $j\ge 1$. This ensures the existence of the limit

$$\rho :=\underset{j\to \infty }{lim}j{\pi }_j\in (0,1].$$

(15)

Let ${\mathsf{\Pi }}_{\rho }$ denote the subset of ${\mathsf{\Pi }}_{+}$ satisfying (13) and (15).

If $\pi \in {\mathsf{\Pi }}_{\rho }$, then $\pi /\rho \in {\mathsf{\Pi }}_1$, and hence, we can, and usually will, assume that the limit (15) equals unity, in which case, the resulting pgf $B(s)$ induces a family of discrete laws whose pgf is

$$B(s;\rho )=\sum _{j=1}^{\infty }{b}_j(\rho )s^j=1-{(1-B(s))}^{\rho },(0<\rho \le 1).$$

(16)

Remark 3. 

The convention in Remark 2 implies the following representation: For $j\ge 1$,

$${\pi }_j=j^{-1}(1-{\theta }_j)\mathrm{where}0\le {\theta }_{j+1}\le {\theta }_j<1.$$

(17)

Thus, choosing simple forms of ${\theta }_j$ provides an easy recipe for constructing pgfs. We illustrate this in the following simple example.

Example 1. 

Let $p\in (0,1)$ and ${\theta }_j=p^j$. Then, $\pi (s)=log[(1-ps)/(1-s)]$ and

$$B(s)=s{\displaystyle \frac{1-p}{1-ps}},$$

corresponding to a shifted geometric law. This is the case $\nu =1$ in §4 of [3], where it is shown that $B(s)$ is the pgf of the LCL of a linear birth-and-death process. See Example 5 below.

Next, we examine the right-hand tail behaviour of the laws $(b_j(\rho ))$. This draws on Theorem 2.3 in [6] although Part (a) is new. We denote with $SV$ the set of measurable functions which are slowly varying at infinity, and with $S{V}_0$ the set of normalised members of $SV$; see [11] for these familiar notions. In addition, for $j=0,1,\dots$, we define the harmonic numbers $H(j)={\sum }_{i=1}^j{i}^{-1}$.

Theorem 4. 

(a) Suppose that $\pi \in {\mathsf{\Pi }}_{+}$ and that $j{\pi }_j\uparrow 1$. Then,

$$1-B(s)\sim (1-s)L\left({\displaystyle \frac{1}{1-s}}\right)(s\uparrow 1),$$

(18)

where $L\in S{V}_0$. The condition (18) is equivalent to

$$\sum _{i=1}^j{\pi }_i-H(j)+logL(j)\to 0.$$

(19)

(b) If $0<\rho \le 1$, then, with $L_{\rho }(x):={\left(L(x)\right)}^{\rho }$,

$$1-B(s;\rho )\sim {(1-s)}^{\rho }{L}_{\rho }(1/(1-s)).$$

This condition is equivalent to

$$\sum _{i>j}{b}_i(\rho )\sim j^{-\rho }{L}_{\rho }(j)/\mathsf{\Gamma }(1-\rho )\mathit{if}\rho <1$$

and

$$\sum _{i=1}^{j}i{b}_i\sim L(j)\mathit{if}\rho =1.$$

Proof. 

Clearly

$${\displaystyle \frac{1-B(s)}{1-s}}=e^{-\pi (s)-log(1-s)}=exp\left(\sum _{j=1}^{\infty }{\theta }_j{s}^j/j\right)=exp(I(x)),$$

where $x=1/(1-s)$, and $I(x)={\sum }_{i=1}^{\infty }{i}^{-1}{\theta }_i{\left(1-1/x\right)}^i$ is defined for $x\ge 1$. Differentiating term-wise yields

$$\epsilon (x):=x{I}^{\prime }(x)=x^{-1}\sum _{i=1}^{\infty }i{\theta }_i{\left(1-1/x\right)}^i.$$

But ${\theta }_i\downarrow 0$, and hence, for small $ϵ>0$, we can choose $i^{\prime }$ such that ${\theta }_i\le ϵ$ if $i>i^{\prime }$. Hence,

$$\epsilon (x)\le O\left(x^{-1}\right)+{\displaystyle \frac{ϵ}{x}}·{\displaystyle \frac{{\left(1-1/x\right)}^{i^{\prime }}}{1-\left(1-1/x\right)}}\le O\left(x^{-1}\right)+ϵ.$$

Hence, ${lim}_{x\to \infty }\epsilon (x)=0$. It follows that

$$L(x)=e^{I(x)}=exp\left({\int }_1^x\epsilon (y)dy/y\right)\in S{V}_0,$$

and (18) follows.

Next, it follows from the definition of ${\theta }_j$ that ${\sum }_{i=1}^j{\pi }_i=H(j)-{\sum }_{i=1}^j{\theta }_i/i$ and $logL(j)=I(j)={\sum }_{i=1}^{\infty }{i}^{-1}{\theta }_i{(1-1/j)}^i$, and hence,

$$\sum _{i=1}^j{\pi }_i-H(j)+logL(j)=-\sum _{i=1}^j{i}^{-1}{\theta }_i\left[1-{(1-1/j)}^i\right]+\sum _{i=j+1}^{\infty }{i}^{-1}{\theta }_i{(1-1/j)}^i.$$

The first sum on the right-hand side equals

$$\sum _{i=1}^j{\theta }_i{\int }_{1-1/j}^1{u}^{i-1}du\le j^{-1}\sum _{i=1}^j{\theta }_i\to 0.$$

Similarly, the second sum is bounded above by

$${\theta }_j\sum _{i>j}{\int }_0^{1-1/j}{u}^{i-1}du={\theta }_j{\int }_0^{1-1/j}{\displaystyle \frac{u^j}{1-u}}du<{\theta }_{j}j{\int }_0^{1-1/j}{u}^{j}du={\theta }_j{\displaystyle \frac{j}{j+1}}{(1-1/j)}^{j+1}\to 0.$$

The estimate (19) follows.

The equivalence of (18) and (19) and the assertions of Part (b) follow from Tauberian theory, as in the proof of Theorem 2.3 in [6]. □

Remark 4. 

As observed in [6], the mean of $(b_j)$ is $m_b=exp\left\{{\sum }_{j=1}^{\infty }{\theta }_j/j\right\}$, and if $m_b<\infty$, then $m_b^{-1}(1-B(s))/(1-s)$ is the pgf of a compound Poisson law.

We deviate from the main theme to observe that if $\rho \in (0,1)$, then the derived pgf $B(s;\rho )=h_{\rho }(B(s))$ is a compounded pgf, because

$$h_{\rho }(s):=1-{(1-s)}^{\rho }=\rho \sum _{j=1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }(j-\rho )}{\mathsf{\Gamma }(1-\rho )}}·{\displaystyle \frac{s^j}{j!}}$$

is obviously the pgf of a positive-integer-valued random variable ($N_{\rho }$, for example) with the Sibuya distribution. Hence $B(s;\rho )$ is the pgf of the random sum ${\sum }_{n=1}^{N\rho }{\mathsf{\Lambda }}_n$, where the summands are independent with pgf B and independent of $N_{\rho }$.

The pgf of ${\mathcal{N}}_{\rho }=N_{\rho }-1$ is

$${\eta }_{\rho }(s)=s^{-1}\left(1-{(1-s)}^{\rho }\right)={\psi }_{\rho }(1-s)$$

(20)

where, for $\vartheta \ge 0$,

$${\psi }_{\rho }(\vartheta )=\left\{\begin{array}{cc}{\displaystyle \frac{1-{\vartheta }^{\rho }}{1-\vartheta }}\hfill & \mathrm{if}\vartheta \ne 1,\hfill \\ \rho \hfill & \mathrm{if}\vartheta =1.\hfill \end{array}\right.$$

(21)

The function ${\psi }_{\rho }$ is known to be the Laplace–Stieltjes transform (LST) of a positive random variable (X, for example),

$$\psi (\vartheta )=E\left(e^{-\vartheta X}\right),$$

(22)

and its distribution is a generalised gamma convolution law (abbreviated to GGC), i.e., it belongs to the proper subset of positive infinitely divisible (infdiv) laws, which is the smallest set containing all gamma laws and closed under convolution and weak limits. A GGC is characterised by the property that its Lévy measure has a density $\ell (x)$ such that $x\ell (x)$ is completely monotone:

$$x\ell (x)={\int }_0^{\infty }{e}^{-xt}U(dt)$$

(23)

where $U(dt)$ is the so-called Thorin measure, a positive measure satisfying

$${\int }_{(0,1)}|logt|U(dt)+{\int }_{[1,\infty )}{t}^{-1}U(dt)<\infty .$$

(24)

The representation (23) implies that $x\ell (x)$ is non-increasing, and hence, any GGC is self-decomposable.

See p. 86 in [13] or p. 415 in [14] for the GGC assertion about ${\psi }_{\rho }$, and these references and [15] for the associated general concepts. In summary, these references show that X has an exponential mixture law with the specific representation $X=\epsilon {\gamma }_{1-\rho }/{\gamma }_{\rho }$, where $\epsilon$ has the standard exponential law and ${\gamma }_a$ denotes a random variable with a standard gamma law with shape parameter $a>0$. The three components of X are independent. This identification of the mixing law for the Sibuya distribution was first reported in [16].

The quickest way to obtain this identification of ${\psi }_{\rho }$ is by observing that $Y:={\gamma }_{\rho }/{\gamma }_{1-\rho }$ has the type-2 beta density

$$f_Y(y)=C_{\rho }^{-1}{\displaystyle \frac{y^{\rho -1}}{1+y}}$$

where the normalising constant $C_{\rho }=B(\rho ,1-\rho )$, a beta function. Hence $X=\epsilon /Y$ has the density

$$f_X(x)={\int }_0^{\infty }y{f}_Y(y)e^{-xy}dy=C_{\rho }^{-1}{\int }_0^{\infty }{\displaystyle \frac{y^{\rho }}{1+y}}{e}^{-xy}dy,$$

the form asserted in [14]. The LST (21) follows by writing $X=\epsilon Z$, where $Z=Y^{-1}$ has a type-2 beta law, and then, with $\vartheta >1$,

$$\begin{array}{ccc}\hfill {\psi }_{\rho }(\vartheta )& =& E\left[{(1+\vartheta Z)}^{-1}\right]=C_{\rho }^{-1}{\int }_0^{\infty }{\displaystyle \frac{z^{-\rho }}{(1+\vartheta z)(1+z)}}dz\hfill \\ & =& {\displaystyle \frac{C_{\rho }^{-1}}{\vartheta -1}}{\int }_0^{\infty }\left({\vartheta }^{\rho +1}{f}_Z(\vartheta z)-f_Z(z)\right)dz.\hfill \end{array}$$

The GGC property directly arises from the evident fact that $f_Y(y)$ is hyperbolically completely monotone, i.e., if $v>0$ and $w=v+v^{-1}$, then, for all $u>0$, the product $f_Y(uv)f_Y(u/v)$ is a completely monotone function of w. The density $f_X(x)$ appears on p. 87 of [13] in a more complicated form than that obtained above using the form $X=\epsilon Z$. The details in that study are omitted with a reference to ‘tedious calculations’.

In what follows, we use properties of the generalised Mittag–Leffler function defined in terms of two parameters, $a>0$ and $b\ge 0$, as follows:

$$E_{a,b}(z)=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\mathsf{\Gamma }(an+b)}}.$$

(25)

When $b=0$, it is understood that $\mathsf{\Gamma }(0)=+\infty$. The original one-parameter version is $E_a(z)=E_{a,1}(z)$, which generalises the exponential function, $E_1(z)=e^z$. We list known properties used in the sequel (p. 210 in [17], or [18]). First, there is an LST identity,

$${\int }_0^{\infty }{e}^{-\vartheta x}{x}^{b-1}{E}_{a,b}\left(x^a\right)={\displaystyle \frac{{\vartheta }^{-b}}{1-{\vartheta }^{-a}}},(\vartheta >1).$$

(26)

An asymptotic form, valid as $x\to \infty$, $a<2$ and integer $n^{\prime }>1$, is

$$a{E}_{a,b}(x^a)=x^{1-b}{e}^x-\sum _{n=1}^{n^{\prime }}{\displaystyle \frac{x^{-an}}{\mathsf{\Gamma }(b-an)}}+O\left(x^{-a(n^{\prime }+1)}\right).$$

(27)

The following result is an (almost) immediate consequence of (26) and (27). In particular, it gives a very different-looking representation of $f_X(x)$, which is not at all evident from the above integral form.

Theorem 5. 

The density function of X specified by (21) and (22) has the representation

$$f_X(x)=x^{-\rho }{E}_{1,1-\rho }(x)-e^x,$$

and its asymptotic expansion as $x\to \infty$ is, if $n^{\prime }\ge 1$,

$$f_X(x)=-\sum _{n=1}^{n^{\prime }}{\displaystyle \frac{x^{-n-\rho }}{\mathsf{\Gamma }(1-\rho -n)}}+O\left(x^{-n^{\prime }-1}\right).$$

Clearly, ${lim}_{\vartheta \to \infty }{\vartheta }^{-1}log{\psi }_{\rho }(\vartheta )=0$, and hence, the cumulant-generating function (cgf) $c_{\rho }(\vartheta ):=-log{\psi }_{\rho }(\vartheta )$ of the infinitely divisible random variable X is, for $\vartheta >1$,

$$c_{\rho }(\vartheta )={\int }_0^{\infty }\left(1-e^{-\vartheta x}\right){\lambda }_{\rho }(dx)=log(\vartheta -1)-log({\vartheta }^{\rho }-1),$$

where ${\lambda }_{\rho }(dx)$ is the Lévy measure of the law of X, which, by definition, satisfies the condition ${\int }_0^{\infty }(x\wedge 1){\lambda }_{\rho }(dx)<\infty$.

Differentiation of the cgf yields the Laplace transform relation

$$c_{\rho }^{\prime }(\vartheta )={\int }_0^{\infty }{e}^{-\vartheta x}x{\lambda }_{\rho }(dx)={\displaystyle \frac{1}{\vartheta -1}}-\rho {\displaystyle \frac{{\vartheta }^{\rho -1}}{{\vartheta }^{\rho }-1}}={\displaystyle \frac{1}{\vartheta -1}}-\rho {\displaystyle \frac{{\vartheta }^{-1}}{1-{\vartheta }^{-\rho }}}.$$

The first assertion of the next theorem follows from the above discussion, (26) and (27). The second assertion follows from Theorem 7 below.

Theorem 6. 

(i) The law of X specified by (21) and (22) is a GGC, and its Lévy measure ${\lambda }_{\rho }(dx)$ is absolutely continuous with density ${\ell }_{\rho }(x)$ given by

$$x{\ell }_{\rho }(x)=e^x-\rho E_{\rho }(x^{\rho })=\sum _{n=2}^{n^{\prime }}{\displaystyle \frac{x^{-\rho n}}{\mathsf{\Gamma }(1-\rho n)}}+O\left(x^{-\rho (n^{\prime }+1)}\right)(n^{\prime }\ge 2,x\to \infty ).$$

(ii) The corresponding Thorin measure is absolutely continuous, $U_{\rho }(dt)=u_{\rho }(t)dt$, where

$$u_{\rho }(t)={\displaystyle \frac{sin\pi \rho }{\pi }}·{\displaystyle \frac{\rho t^{\rho -1}}{1+t^{\rho }cos\pi \rho +t^{2\rho }}},(t>0).$$

(28)

A function similar to $u_{\rho }(t)$ arises as the density function of a quotient of independent positive stable random variables. Specifically, $u_{\rho }(t)/(1-\rho )$ is the density of ${\left(S_{1-\rho }/S_{1-\rho }^{\prime }\right)}^{{\rho }^{-1}-1}$, where the components of the quotient are independent and they have a positive stable law with index $1-\rho$.

The fact that X is self-decomposable implies that it has the ‘in law’ stochastic integral representation

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

where $(L_t:t\ge 0)$ is a subordinator. Its cgf $k_{\rho }(\vartheta )$ (i.e., $E\left(\left.e^{-\vartheta L_t}\right|L_0=0\right)=e^{-t{k}_{\rho }(\vartheta )}$) is related to $c_{\rho }(\vartheta )$ by ${\vartheta }^{-1}{k}_{\rho }(\vartheta )=c_{\rho }^{\prime }(\vartheta )$. See [19] for these connections. The left-hand side of this identity is the Laplace transform of the right-hand tail ${\overline{\mathsf{\Lambda }}}_{\rho }(x)$ of the Lévy measure of the subordinator. Hence, we have the identification ${\overline{\mathsf{\Lambda }}}_{\rho }(x)=x{\ell }_{\rho }(x)$. In particular, ${\overline{\mathsf{\Lambda }}}_{\rho }(x)$ is the Laplace transform of the Thorin density $u_{\rho }(t)$.

The pgf ${\eta }_{\rho }(s)$ (20) has a canonical representation as the pgf of a generalised negative-binomial convolution (i.e., a GGC mixture of Poisson laws; see p. 388 in [14]),

$${\eta }_{\rho }(s)=e^{-r_{\rho }(1-Q_{\rho }(s))}=exp\left[-{\int }_0^{1}log{\displaystyle \frac{1-ps}{1-p}}{v}_{\rho }(p)dp\right],$$

where $v_{\rho }(p)=p^{-2}{u}_{\rho }\left(p^{-1}-1\right)$. Normalising the Poisson jump-size pgf as $Q_{\rho }(0)=0$, i.e., $Q_{\rho }(s)={\sum }_{j=1}^{\infty }{q}_j(\rho )s^j$, we have the evaluations

$$r_{\rho }=-log\rho =-{\int }_0^{1}log(1-p)v_{\rho }(p)dp\mathrm{and}{q}_j(\rho )={(r_{\rho }j)}^{-1}{\int }_0^1{p}^j{v}_{\rho }(p)dp.$$

The author was initially led to Theorems 5 and 6 through finding Entry 11 in the catalogue of complete Bernstein functions recorded in [15]. This asserts that if $0<\alpha <\beta <1$, then

$$c(\vartheta )=\left\{\begin{array}{cc}{\displaystyle \frac{1-{\vartheta }^{\beta }}{1-{\vartheta }^{\alpha }}}-1\hfill & \mathrm{if}\vartheta \ne 1,\hfill \\ \beta /\alpha -1\hfill & \mathrm{if}\vartheta =1\hfill \end{array}\right.$$

(29)

is complete-Bernstein, meaning that it is the cgf of a positive infdiv law whose Lévy measure has a completely monotone density. In fact, the source for this identification [20] allows the parameter range $0<\alpha <\beta \le 1$. It follows that there is a subordinator $(V_t:t\ge 0)$ such that $V_0=0$ and $E(e^{-\vartheta V_t})=e^{-tc(\vartheta })$. Furthermore, if $\epsilon$ has the standard exponential law and is independent of $(V_t)$, then the exponentially stopped version $W=V_{\epsilon }$ has the LST

$$\omega (\vartheta )=E\left(e^{-\vartheta W}\right)={\left(1+c(\vartheta )\right)}^{-1}={\displaystyle \frac{1-{\vartheta }^{\alpha }}{1-{\vartheta }^{\beta }}}$$

(30)

and it has an infdiv law. The law of X above is recovered by setting $\alpha =\rho$ and $\beta =1$.

The next theorem collects the infdiv properties of W.

Theorem 7. 

If $0<\alpha <\beta \le 1$, then the law of W specified by (30) is a GGC. Its pdf is

$$f_W(x)=x^{-(\alpha +1-\beta )}{E}_{\beta ,\beta -\alpha }(x^{\beta })-x^{\beta -1}{E}_{\beta ,0}(x^{\beta })$$

(31)

and

$$f_W(x)\sim {\displaystyle \frac{\alpha }{\beta \mathsf{\Gamma }(1-\alpha )}}{x}^{-1-\alpha }(x\to \infty ).$$

The Lévy measure $J(dx)$ of W is absolutely continuous with the density $j(x)$ given by

$$xj(x)=\beta E_{\beta }(x^{\beta })-\alpha E_{\alpha }(x^{\alpha })$$

(32)

and the corresponding Thorin measure is absolutely continuous with density (recalling (28))

$$u(t)=u_{\alpha }(t)-u_{\beta }(t).$$

(33)

Proof. 

We observe first that the case $\beta <1$ reduces to the case $\beta =1$, as can be seen by defining $\rho =\alpha /\beta$ and observing that $\omega (\vartheta )={\psi }_{\rho }\left({\vartheta }^{\beta }\right)$. This implies the representation $W\stackrel{L}{=}{X}^{1/\beta }{S}_{\beta }$, where X is as above, and $S_{\beta }$ is independent of X and has the positive stable law with index $\beta$, with the understanding that $S_1\stackrel{L}{=}1$. Alternatively, $W\stackrel{L}{=}{S}_{\beta }(X)$, where $(S_{\beta }(t):t\ge 0)$ is the stable subordinator with $S_{\beta }(1)\stackrel{L}{=}{S}_{\beta }$. In particular, it follows from Theorem 3.3.2 in [13] that W has a GGC law.

Next, the density (31) follows by rewriting (30) as

$$\omega (\vartheta )={\displaystyle \frac{{\vartheta }^{-(\beta -\alpha )}-{\vartheta }^{-\beta }}{1-{\vartheta }^{-\beta }}},$$

and referring to (26). The asymptotic estimate follows from (27).

The identity (32) follows by writing $\omega (\vartheta )=e^{-k(\vartheta })$ and observing that

$${\int }_0^{\infty }{e}^{-\vartheta x}xJ(dx)=k^{\prime }(\vartheta )=\alpha {\displaystyle \frac{{\vartheta }^{\alpha -1}}{1-{\vartheta }^{\alpha }}}-\beta {\displaystyle \frac{{\vartheta }^{\beta -1}}{1-{\vartheta }^{\beta }}}$$

and referring, again, to (26).

Finally, it follows from Theorem 3.1.4 in [13] that the Thorin density $u(t)$ exists and that $u(t)=-{\pi }^{-1}\Im k^{\prime }(-t)$, where $\Im (·)$ denotes the imaginary part of $(·)$. But

$$\begin{array}{ccc}\hfill \Im \left(-{\displaystyle \frac{{(-t)}^{\alpha -1}}{1-{(-t)}^{\alpha }}}\right)& =& t^{-1}\Im {\displaystyle \frac{t^{\alpha }{e}^{i\pi \alpha }}{1-t^{\alpha }{e}^{i\pi \alpha }}}\hfill \\ & =& t^{-1}\Im {\displaystyle \frac{t^{\alpha }\left(e^{i\pi \alpha }-1\right)}{\left(1-t^{\alpha }{e}^{i\pi \alpha }\right)\left(1-t^{\alpha }{e}^{-i\pi \alpha }\right)}}=(\pi /\alpha )u_{\alpha }(t).\hfill \end{array}$$

The identity (33) follows □

Remark 5. 

The identity (28) follows from (33) with $\alpha =\rho$ and $\beta =1$, and we observe that $u_1(t)\equiv 0$.

Returning to the complete Bernstein function (29), denote its (completely monotone) Lévy density as $m(x)$, and let $\overline{M}(x)={\int }_x^{\infty }m(y)dy$, whose Laplace transform is

$${\vartheta }^{-1}c(\vartheta )={\displaystyle \frac{{\vartheta }^{-(1-\beta +\alpha )}-{\vartheta }^{-1}}{1-{\vartheta }^{-\alpha }}}.$$

It follows from (26) that

$$\overline{M}(x)=x^{-(\beta -\alpha )}{E}_{\alpha ,\alpha +1-\beta }(x^{\alpha })-E_{\alpha }(x^{\alpha }).$$

The derivative identity

$${\displaystyle \frac{d}{dx}}{x}^{b-1}{E}_{a,b}(x^a)=x^{b-2}{E}_{a,b-1}(x^a)$$

yields the explicit expression

$$m(x)=x^{-1}{E}_{\alpha ,0}(x^{\alpha })-x^{-(\beta +1-\alpha )}{E}_{\alpha ,\alpha -\beta }(x^{\alpha }),$$

with the understanding that any term in the series expansion (25) for which the gamma function denominator has a non-positive integer argument contributes zero to the sum. It follows from (27) that

$$m(x)\sim {\left(\mathsf{\Gamma }(1-\alpha )x^{1+\alpha }\right)}^{-1}.$$

The complete Bernstein function $c(\vartheta )$ has a Stieltjes transform representation

$$c(\vartheta )={\int }_0^{\infty }{\displaystyle \frac{\vartheta }{t+\vartheta }}\sigma (dt)$$

for a certain absolutely continuous measure whose density is listed in [15] and derived in [20]. On the other hand, the Lévy density has a Laplace–Stieltjes representation $m(x)={\int }_0^{\infty }{e}^{-xt}V(dt)$, implying that

$$c(\vartheta )={\int }_0^{\infty }{\displaystyle \frac{\vartheta }{t(t+\vartheta )}}V(dt).$$

Hence, $V(dt)=t\sigma (dt)$, and, referring to the table entry in [15], $V(dt)=v(t)dt$, where

$$v(t)={\displaystyle \frac{1}{\pi \alpha }}·{\displaystyle \frac{t^{\alpha }sin\pi \alpha -t^{\beta }sin\pi \beta +t^{\alpha +\beta }sin\pi (\alpha +\beta )}{1-2t^{\alpha }cos\pi \alpha +t^{2\alpha }}}.$$

(34)

We are now able to state the following result.

Theorem 8. 

The cgf (29) is Thorin–Bernstein, and its Thorin measure has the density $v^{\prime }(t)$.

Proof. 

The function $v(t)$ is differentiable, and $v(0)=0$. Hence,

$$m(x)={\int }_0^{\infty }{e}^{-xy}v(y)dy={\int }_0^{\infty }{e}^{-xy}{\int }_0^y{v}^{\prime }(t)dtdy,$$

i.e.,

$$xm(x)={\int }_0^{\infty }{e}^{-xt}{v}^{\prime }(t)dt.$$

Hence, a necessary and sufficient condition for $c(\vartheta )$ to be Thorin–Bernstein is that $v^{\prime }(t)\ge 0$ in $(0,\infty )$.

It suffices to consider only the case $\beta =1$, for if $\beta <1$, then defining $\rho =\alpha /\beta$, we have $c(\vartheta )={\widehat{c}}_{\rho }({\vartheta }^{\beta })$, where

$${\widehat{c}}_{\rho }(\vartheta )={\displaystyle \frac{1-\vartheta }{1-{\vartheta }^{\rho }}},$$

and, appealing to Theorem 3.3.2 in [13], we see that proving ${\widehat{c}}_{\rho }(\vartheta )$ is Thorin–Bernstein will prove it for $c(\vartheta )$.

The corresponding form of (34) is

$$v_{\rho }(t)={\displaystyle \frac{sin\pi \rho }{\pi \rho }}·{\displaystyle \frac{t^{\rho }+t^{1+\rho }}{1-2t^{\rho }cos\pi \rho +t^{2\rho }}}.$$

Finally, setting $z=t^{\rho }$, it follows that $v_{\rho }^{\prime }(t)\ge 0$ if and only if $R^{\prime }(z)\ge 0$, where

$$R(z)={\displaystyle \frac{z+z^{1+\zeta }}{1-2cz+z^2}},$$

$\zeta =1/\rho >1$ and $c=cos\pi \alpha$. Note that the denominator is positive.

Carrying out the differentiation yields $R^{\prime }(z)=N(z)/{\left(1-2cz+z^2\right)}^2$, where, suppressing the subscript $\rho$,

$$N(z)=1-z^2+z^{\zeta }Q(z)$$

and

$$Q(z)=1+\zeta -2c\zeta z+(\zeta -1)z^2,$$

a convex function satisfying $Q(0)=1+\zeta >0$ and $Q^{\prime }(0)=-2c\zeta$. We look at three cases.

First, if $\rho =\frac{1}{2}$, then $N(z)=1+2z^2+z^4>0$. Next, if $\rho >\frac{1}{2}$, then $\gamma =-c>0$ and $Q^{\prime }(0)>0$. Hence, $Q(z)$ is convex increasing from $Q(0)$ and $N(z)>0$ if $z\in [0,1]$. Noting that $1<\zeta <2$, it follows for $z>1$ that

$$N(z)>1-z^2+zQ(z)=K(z):=1+(1+\zeta )z+(2\gamma \zeta -1)z^2+(\zeta -1)z^3.$$

But $K(1)=1$,

$$K^{\prime }(1)=4\gamma \zeta +2(\zeta -1)>0$$

and

$$K^{\prime \prime }(z)=2(2\gamma \zeta -1)+6(\zeta -1)z\ge 2\zeta [2\gamma +3-4\rho ],$$

remembering that $z\ge 1$.

The term in square brackets, $q(\rho )=3-2cos\pi \rho -4\rho$, satisfies $q(\frac{1}{2})=q(1)=1$, $q^{\prime }(\frac{1}{2})=2(\pi -1)>0$, $q^{\prime }(1)=-4$ and $q^{\prime \prime }(\rho )=2{\pi }^{2}cos\pi \rho <0$. Hence, $q(·)$ is concave increasing and then decreasing, and $q(\rho )>1$, if $\rho \in (\frac{1}{2},1)$. Hence, $N(z)>0$ for all $z>0$.

Next, let $0<\rho <\frac{1}{2}$, implying that $\zeta >2$. The discriminant of $Q(z)$ is

$$D=4c^2{\zeta }^2-4({\zeta }^2-1)=4\left[1-{\pi }^2{\left({\displaystyle \frac{sin\pi \rho }{\pi \rho }}\right)}^2\right]<0$$

because ${\varphi }^{-1}sin\varphi$ decreases from unity to $2/\pi$ across $[0,\pi /2]$. Hence, $Q(z)>0$ if $z\ge 0$, and $N(z)>0$ if $0\le z\le 1$.

If $z\ge 1$, then

$$N(z)\ge 1-z^2+Q(z)=\tilde{Q}(z):=(\zeta -2)z^2-2c\zeta z+2+\zeta .$$

Obviously, $\tilde{Q}(·)$ is convex in $[1,\infty )$ and $\tilde{Q}(1)=2\zeta (1-c)>0$. In addition, $\tilde{Q}(·)$ has a single minimum at $z_m={(1-2\rho )}^{-1}cos\pi \rho$. The numerical computation shows that $z_m$ increases from unity at $\rho =0$ to $\pi /2=1.5708$ at $\rho =\frac{1}{2}$. In addition, ${\tilde{Q}}^{\prime }(1)=2(\zeta -2)(1-z_m)<0$. Further numerical calculation shows that $\tilde{Q}(z_m)$ increases from 0 at $\rho =0$ to 2 at $\rho =\frac{1}{2}$. Hence, $N(z)>0$ if $z\ge 1$.

It follows in all cases that $v^{\prime }(t)>0$, as desired. Finally, the integrability condition (24) with $U(dt)=v^{\prime }(t)dt$ is equivalent to

$${\int }_0^1{t}^{-1}v(t)dt+{\int }_1^{\infty }{t}^{-2}v(t)dt<\infty .$$

This condition is satisfied because $v(t)\sim (sin\pi \alpha )t^{\alpha }$ as $t\downarrow 0$ and $v(t)\sim (sin\pi (\beta -\alpha ))t^{\beta -\alpha }$ as $t\to \infty$. It follows that $v^{\prime }(t)$ is the density of a Thorin measure. □

4. The Subcritical Case

In this section, $K=1-m>0$, in which case, it follows from Theorem 1 that the stationary measure satisfies $j{\pi }_j\uparrow 1$. Hence, the parts of Theorem 4 with $\rho =1$ apply to LCLs. In relation to Remark 2, we recall that that $m_B:=B^{\prime }(1-)<\infty$ if and only if the so-called $xlogx$ condition holds, i.e., ${\sum }_{j=1}^{\infty }(jlogj)p_j<\infty$.

A further general point is that if $B\in LCL$, then the derived pgfs $B(s;\rho )$ arise as the pgfs of LCLs in the following extended sense. Let $\nu$ denote an arbitrary initial law, ${\nu }_i=P(Z_0=i)$ with $i\ge 1$. If the limits

$${\tilde{b}}_j=\underset{t\to \infty }{lim}{P}_{\nu }(Z_t=j|T>t)$$

exist and not all of them are zero, then there is a constant $\rho \in (0,1]$ such that $\tilde{B}(s)={\sum }_{j=1}^{\infty }{\tilde{b}}_j{s}^j$ solves the functional equation

$$1-\tilde{B}(F(s,t))=e^{-\rho \mu t}(1-\tilde{B}(s)).$$

This follows from the analogous result for the GWP, p. 65 in [21]. Differentiating with respect to t using the backward equation in the form $\partial F(s,t)/\partial t=a(f(F(s,t))-F(s,t))$ leads to the first equality in the identity

$${\displaystyle \frac{{\tilde{B}}^{\prime }(s)}{1-\tilde{B}(s)}}=\rho {\displaystyle \frac{1-m}{f(s)-s}}=\rho {\displaystyle \frac{B^{\prime }(s)}{1-B(s)}},$$

and the second equality results from (3) and (1). Hence, $log(1-\tilde{B}(s))=\rho log(1-B(s))$, i.e., $\tilde{B}(s)=B(s;\rho )=h_{\rho }(B(s))$. Thus, $({\tilde{b}}_j)$ is a non-defective conditional limiting law in the above wider sense. In addition, the limits ${\tilde{b}}_j$ exist if and only if for some $\rho \in (0,1]$, the initial law satisfies

$$1-\nu (s)={(1-s)}^{\rho }{L}_{\rho }\left({\displaystyle \frac{1}{1-s}}\right),L_{\rho }\in S{V}_{\infty },$$

i.e., it is attracted to the positive stable law with index $\rho$.

With our chosen scaling, it follows from Theorem 1 that $\pi \in SM$ satisfies (13) and (14). Conversely, if $\pi \in {\mathsf{\Pi }}_{+}$ satisfies these conditions and if the resulting pgf $B(s)$ satisfies the conditions of the next theorem, then $\pi \in SM$. The next theorem is simply a recasting of Theorem 2.1 in [3] expressed in a stronger form.

Theorem 9. 

Suppose that $B(s)={\sum }_{j=1}^{\infty }{b}_j{s}^j$ is a pgf satisfying $b_1>0$. Then, the function $\beta (s)=(1-B(s))/B^{\prime }(s)$ has a power series expansion converging for $|s|<1$, $\beta (s)={\sum }_{j=0}^{\infty }{\beta }_j{s}^j$. Let $m\in (0,1)$ be a constant satisfying

$$m_l:=1+1/{\beta }_1\le m<1$$

(35)

and suppose that ${\beta }_j\ge 0$ for all $j\ge 2$. Then,

$$f(s)=s+(1-m)\beta (s)\in SC$$

(36)

and $B(s)$ is the pgf of the corresponding LCL.

Conversely, if $f\in SC$, then it is related to the pgf $B(s)$ of its LCL by the identity (36) with $\beta (s)$ as above, and (35) holds.

Remark 6. 

The condition (35) is equivalent to $0\le p_1<1$. In addition,

$$m_l={\displaystyle \frac{2b_2/b_1^2}{1+2b_2/b_1^2}}.$$

There is a very obvious corollary of Theorems 3 and 9, viz., if (13) and (14) both hold and if $B(s)$ satisfies the conditions of Theorem 9, then $\pi \in S{M}_f$, with f as in (36) and (35). Not surprisingly, there is a gap in that the distributions derived from Theorem 3 may not be the LCL of a MBP. One example suffices to show this.

Example 2. 

Given constants $\lambda \in (0,1)$ and $c\in [\lambda ,1]$, let ${\theta }_1=1-\lambda$ and ${\theta }_j=1-c$ if $j\ge 2$. Then, $\pi (s)=\lambda s-clog(1-s)$, and $B(s;c)=1-{(1-s)}^c{e}^{\lambda s}$ is a pgf. It follows that $\beta (s)=(1-s)/(c-\lambda +\lambda s)$, and it is evident that the coefficients ${\beta }_j$ alternate in sign. Hence $\pi \notin SM$.

The next two examples complete the details for two examples of LCLs exhibited in [3] by identifying the stationary measure.

Example 3. 

Let $\lambda >0$, $\theta =\lambda e^{-\lambda }$, and $C(z)$ denote the Cayley tree counting function, i.e., the positive-valued solution of the functional equation $C(z)=z{e}^{C(z)}$. This function is analytic in a disc of radius $e^{-1}$. It is shown in [3] that if $\lambda \in (0,1)$, then $B_{\lambda }(s):={\lambda }^{-1}C(\theta s)$ is the pgf of a Borel law and it is an LCL. The gf of the corresponding stationary measure is

$${\pi }_{\lambda }(s)=-log\left(1-{\lambda }^{-1}C(\theta s)\right)=h(C(\theta s)),$$

where $h(z)=-log(1-z/\lambda )$. Now, $h^{\prime }(z)={(\lambda -z)}^{-1}$ ($|z|<\lambda$) and

$${\displaystyle \frac{d^i}{d{z}^i}}{(\lambda -z)}^{-1}=i!{(\lambda -z)}^{-i-1}.$$

Using the notation $D_0^i(·)=(d^i/d{z}^i)(·){|}_{z=0}$, it follows from the Leibniz rule that

$$d_j:=D_0^{j-1}\left(h^{\prime }(z)e^{jz}\right)=\sum _{i=0}^{j-1}i!{\lambda }^{-i-1}{j}^{j-1-i}.$$

Lagrange reversion of the series thus leads to the evaluation

$${\pi }_j=d_j{\displaystyle \frac{{\theta }^j}{j!}}(j\ge 1).$$

It is not at all obvious from mere inspection that the conditions of Theorem 3 are satisfied, but they are.

Example 4. 

Let $p\in (0,1)$ and $\ell =-log(1-p)$. It is shown in [3] that the log-series law with pgf $B(s)=-{\ell }^{-1}log(1-ps)$ is the LCL generated by the offspring laws

$$p_0=(1-m){\displaystyle \frac{\ell }{p}},p_1=1-(1-m)(1+\ell )\mathrm{and}{p}_j=(1-m){\displaystyle \frac{p^{j-1}}{j(j-1)}}(j\ge 2),$$

provided that

$$m_l={\displaystyle \frac{\ell }{1+\ell }}\le m<1.$$

A calculation yields

$${\pi }^{\prime }(s)={\displaystyle \frac{B^{\prime }(s)}{1-B(s)}}={\displaystyle \frac{p}{1-ps}}·{\left(\ell +log(1-ps)\right)}^{-1}.$$

(37)

Observe that

$${\left[\ell +log(1-ps)\right]}^{-1}={\int }_0^{\infty }{e}^{-t\ell }{(1-ps)}^{-t}dt=\sum _{j=0}^{\infty }{\xi }_j{(ps)}^j$$

and expand ${(1-ps)}^t$ to see that

$${\xi }_j={\displaystyle \frac{1}{j!}}{\int }_0^{\infty }{(1-p)}^t{\displaystyle \frac{\mathsf{\Gamma }(j+t)}{\mathsf{\Gamma }(t)}}dt.$$

It follows that the stationary measure is given by

$$(j+1){\pi }_{j+1}=p^{j+1}\sum _{i=0}^j{\xi }_i.$$

The last integral $I_j$ can be interpreted in terms of a moment of an exponentially stopped gamma process, $I_j={\ell }^{-1}E({\gamma }_{\tau }^j)$, where $({\gamma }_t:t\ge 0)$ is a standard gamma process (i.e., it is a subordinator, and ${\gamma }_1\sim Exp(1)$) and $\tau \sim Exp(\ell )$ is independent of $({\gamma }_t)$.

Theorem 9 expresses the conditions on a pgf $B(s)$ ensuring it is that of an LCL. The following result is an equivalent version which imposes conditions on a sequence $\pi \in {\mathsf{\Pi }}_1$. Extend the associated $\theta$-sequence by defining ${\theta }_0=1$, thus ensuring that $j{\pi }_j=0$ when $j=0$. Next, for $j\ge 0$, define ${\psi }_j={\theta }_j-{\theta }_{j-1}$ and observe that $({\psi }_j)$ is a pmf whose pgf is $\psi (s)=(1-s){\pi }^{\prime }(s)=(1-s)/\beta (s)$. In addition, ${\psi }_0={\pi }_1$ and ${\psi }_1={\pi }_1-2{\pi }_2$, implying that $2{\pi }_2={\psi }_0-{\psi }_1\ge 0$.

The following result expresses the conditions of Theorem 9 at a more fundamental level, i.e., in terms of a putative stationary measure. Theorem 10 may also be seen as a version of Theorem 2 restricted to the subcritical case.

Theorem 10. 

If $\pi \in {\mathsf{\Pi }}_1$ and $m\in (0,1)$, then the function

$$f(s)=s+(1-m)/{\pi }^{\prime }(s)\in SC$$

if and only if

$$m_l=1-{\displaystyle \frac{{\pi }_0^2}{2{\pi }_2}}\le m<1$$

and the coefficients in the power series $1/{\pi }^{\prime }(s)={\sum }_{j=0}^{\infty }{q}_j{s}^j$ satisfy $q_j\ge 0$ when $j\ge 2$.

Equivalently, suppose that $({\psi }_j:j\ge 0)$ is a pmf with pgf $\psi (s)$ and satisfying ${\psi }_0\in (0,1)$. Then,

$$f(s)=s+(1-m)(1-s)/\psi (s)\in SC$$

if and only if

$$m_l=1-{\displaystyle \frac{{\psi }_0^2}{{\psi }_0+{\psi }_1}}\le m<1$$

(38)

and the coefficients of the power series $1/\psi (s)={\sum }_{j=0}^{\infty }{c}_j{s}^j$ satisfy $c_{j+1}\ge c_j$ for $j\ge 1$.

In addition,

$$p_0=(1-m)/{\psi }_0,p_1=1-(1-m)(c_0-c_1)and{p}_j=(1-m)(c_j-c_{j-1})(j\ge 2).$$

(39)

It seems that, for some examples, it is a little easier to check the conditions of Theorem 10. Revisiting Example 4, it follows from (37) that

$${\displaystyle \frac{p}{\psi (s)}}={\displaystyle \frac{1-ps}{1-s}}\left(\ell +log(1-ps)\right)$$

and, for $j\ge 1$,

$$p{c}_j=-j^{-1}{p}^{j+1}+(1-p)\left[\ell -\sum _{i=1}^j{p}^i/i\right]=-\sum _{i=j+1}^{\infty }{\displaystyle \frac{p^i}{i(i-1)}}\uparrow 0.$$

In the next example, we choose a simple form for the sequence $({\theta }_j)$ to generalise the case of a geometric offspring law.

Example 5. 

The geometric offspring law $f(s)={(1+m-ms)}^{-1}$ is examined in [8] where, with different notation, it is shown that the corresponding LCL has the pgf

$$B(s)=1-{\displaystyle \frac{1-s}{{(1-ms)}^m}}.$$

The evaluation of $\pi (s)$ yields $j{\pi }_j=1-m^{j+1}$, i.e., ${\theta }_j=m\times m^j$.

We explore the generalisation ${\theta }_j=\alpha p^j$, where $p\in (0,1)$ and $\alpha >0$, i.e., $j{\pi }_j=1-\alpha p^j$. Requiring that ${\pi }_1>0$, we assume that $\alpha p<1$, a condition that is necessary and sufficient for $\pi \in {\mathsf{\Pi }}_1$. Hence,

$$\pi (s)=-log(1-s)+\alpha log(1-ps),$$

(40)

implying that

$$B(s)=1-{\displaystyle \frac{1-s}{{(1-ps)}^{\alpha }}}$$

(41)

is a pgf. In addition, $b_1={\pi }_1>0$, as required. The expansion

$${(1-ps)}^{-\alpha }=\sum _{j=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }(\alpha +j)}{j!\mathsf{\Gamma }(\alpha )}}{(ps)}^j,$$

(42)

leads to the evaluations

$$b_1=1-\alpha p\mathrm{and}{b}_j={\displaystyle \frac{\mathsf{\Gamma }(\alpha +j-1)}{(j-1)!\mathsf{\Gamma }(\alpha )}}{p}^{j-1}\left(1-p+{\displaystyle \frac{(1-\alpha )p}{j}}\right),(j\ge 2).$$

(43)

Computing ${\pi }^{\prime }(s)$ leads to the evaluation

$$\psi (s)={\displaystyle \frac{1-\alpha p-(1-\alpha )ps}{1-ps}}$$

whence, for $j\ge 1$,

$$(1-\alpha p)c_j=(\varrho -p){\rho }^{j-1}\mathrm{where}\varrho ={\displaystyle \frac{(1-\alpha )p}{1-\alpha p}}.$$

The coefficients $c_j$ alternate in sign if $\alpha >1$, but not if $\alpha \le 1$. In this case, $0\le \varrho <1$ and

$$\varrho -p=-{\displaystyle \frac{\alpha p(1-\alpha )}{1-\alpha p}}\le 0.$$

We thus see that the conditions of Theorem 10 are satisfied only if $\alpha \le 1$. Assuming this and observing that ${\psi }_0=1-\alpha p$ and ${\psi }_0+{\psi }_1=1-\alpha p^2$, the condition (38) is

$$1-{\displaystyle \frac{{(1-\alpha p)}^2}{1-\alpha p^2}}\le m<1.$$

(44)

Referring to (41), we conclude that $B(s)\in LCL$ if and only if $p\in (0,1)$, $\alpha \in (0,1]$ and (44) holds. It follows from (39) that the corresponding family of offspring laws is

$$p_0={\displaystyle \frac{1-m}{1-\alpha p}},p_1=1-(1-m){\displaystyle \frac{1-\alpha p^2}{{(1-\alpha p)}^2}}$$

and, for $j\ge 2$,

$$p_j={\displaystyle \frac{1-m}{1-\alpha p}}(1-\varrho )(p-\varrho ){\varrho }^{j-2}.$$

The boundary case $\alpha =1$ entails $\varrho =0$ and, hence, (44) becomes

$${\displaystyle \frac{2p}{1+p}}\le m<1.$$

and $p_j=0$ if $j>2$, corresponding to a family of linear birth–death processes generalised in the sense that $p_1>0$ if $m>m_l$.

In the case $\alpha =p=m$, the geometric offspring law, it is shown in [8] that the inverse $A(y)$ of the function $b(s)=1-B(1-s)$ can be expressed in terms of a so-called generalised Wright function. The point of this is that, if such functional inversion is possible, then it follows from (4) that the population size pgf has an ‘explicit’ representation:

$$F(s,t)=1-A[(1-B(s))e^{-\mu t}].$$

The Wright function, as defined in [8], is better viewed as a four-parameter extension of the confluent hypergeometric function $M(c,d,y)$ in that the definition

$$W(\gamma ,c;\delta ,d;y)=\sum _{k=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }(\gamma k+c)}{\mathsf{\Gamma }(\delta k+d)}}·{\displaystyle \frac{y^k}{k!}}$$

(45)

specialises to $M(c,d,y)=(\mathsf{\Gamma }(c)/\mathsf{\Gamma }(d))W(1,c;1,d;y)$. The actual Wright function $\varphi (\delta ,d;y)$ is the case $\gamma =0$ and $c=1$; see p. 211 in [17].

It follows from (41) that

$$b(s)={\displaystyle \frac{s}{{(1-p+ps)}^{\alpha }}}$$

and hence the inverse $A(y)$ satisfies the functional equation $A(y)=y{(1-p+pA(y))}^{\alpha }$. Lagrange reversion (or the binomial theorem) yields the expansion

$$A(y)=y\sum _{j=0}^{\infty }{a}_{j+1}{\displaystyle \frac{y^j}{(j+1)!}}$$

(46)

where

$$a_{j+1}=D_0^j{(1-p+ps)}^{\alpha (j+1)}=p^j{(1-p)}^{-j(1-\alpha )+\alpha }{\displaystyle \frac{\mathsf{\Gamma }(\alpha j+1+\alpha )}{\mathsf{\Gamma }(1+\alpha -(1-\alpha )j)}}.$$

(47)

The signs of the denominator gamma functions oscillate as j increases. In addition, if $\alpha$ is rational, then $1+\alpha -(1-\alpha )j$ can take non-positive integral values. The corresponding coefficient $a_{j+1}=0$. The numerator gamma function equals $(j+1)\mathsf{\Gamma }(\alpha +\alpha j)$, so, substituting this into (47) and comparing the resulting form of (46) with (45), we obtain

$$A(y)={(1-p)}^{\alpha }yW\left(\alpha ,\alpha ;-(1-\alpha ),\alpha +1;{\displaystyle \frac{py}{(1-p^{1-\alpha }}}\right).$$

This evaluation agrees with the evaluation of ${\mathcal{A}}^{-1}(x)$ in [8] (where $\alpha =p=m$; see p. 363) after correcting a minor error: the factor $1-m$ should be ${(1-m)}^m$. The error arises in going from line -10 to -6:

$${(1-m)}^{mk}{\left({\displaystyle \frac{m}{1-m}}\right)}^{k-1}={(1-m)}^m{\left[{(1-m)}^m·{\displaystyle \frac{m}{1-m}}\right]}^{k-1}={(1-m)}^m{\left[{\displaystyle \frac{m}{{(1-m)}^{1-m}}}\right]}^{k-1}.$$

By following the line of argument in [8], one evaluates the radius of convergence of the series (46) as $p^{-1}{\alpha }^{-\alpha }{[(1-p)/(1-\alpha )]}^{1-\alpha }$.

5. Critical Case: Conditioning on Time of Extinction

In this section, we let $m=1$ and write $\mathcal{U}(s)=a(f(s)-s)$. The pgf $F(s,t)$ satisfies the gf form of the backward Kolmogorov system:

$${\displaystyle \frac{\partial F}{\partial t}}(s,t)=\mathcal{U}(F(s,t)).$$

(48)

Choosing $K=1$ in (1), the integrated version of (48) can be expressed as

$$\pi (F(s,t))=\pi (s)+at.$$

(49)

Setting $s=0$ and observing that $P_1(T\le t)=p_{i0}(t)=F^i(t)$, where $F(t)=F(0,t)$, we have $\pi (F(t))=at$.

In this section we pursue the MBP version of the critical GWP conditional limit (7).

Theorem 11. 

If $m=1$, and $\tau >0$ is a fixed number, then

$$\begin{array}{ccc}\hfill L_j(\tau )& =& \underset{y\to 0}{lim}\underset{t\to \infty }{lim}{P}_i(Z_t=j|t+\tau <T\le t+\tau +y)\hfill \\ & =& \underset{t\to \infty }{lim}\underset{y\to 0}{lim}{P}_i(Z_t=j|t+\tau <T\le t+\tau +y)\hfill \\ & =& {\displaystyle \frac{j{\pi }_j{(F(\tau ))}^{j-1}}{{\pi }^{\prime }(F(\tau ))}},(j=1,2,\dots ).\hfill \end{array}$$

Proof. 

The first double limit follows from Theorem 7.2 in [5]: If $\mathcal{B}$ is a Borel subset of the positive reals with Lebesgue measure $|\mathcal{B}|\in (0,\infty )$, then

$$\underset{t\to \infty }{lim}{P}_i(Z_t=j|T\in t+\mathcal{B})={\displaystyle \frac{{\pi }_j}{a|\mathcal{B}|}}{\int }_{{\mathcal{B}}_F}d{u}^j,$$

(50)

where ${\mathcal{B}}_F=\{u=F(t):t\in \mathcal{B}\}$, i.e., ${\mathcal{B}}_F$ is the image set $F(\mathcal{B})$.

If $y>0$ and $\mathcal{B}=(\tau ,\tau +y]$, then $|\mathcal{B}|=y$ and ${\mathcal{B}}_F=(F(\tau ),F(\tau +y)]$. The integral in (50) has the evaluation $F^j(\tau +y)-F^j(\tau )$. Hence,

$$\underset{t\to \infty }{lim}{P}_i(Z_t=j|t+\tau <T\le t+\tau +y)={\displaystyle \frac{{\pi }_j}{ay}}\left([F^j(\tau +y)-F^j(\tau )\right].$$

It follows from (48) that

$$\underset{y\to 0}{lim}{y}^{-1}\left[F^j(\tau +y)-F^j(\tau )\right]=j{F}^{\prime }(\tau )F^{j-1}(\tau )=j\mathcal{U}(F(\tau ))F^{j-1}(\tau ).$$

Hence, the first double limit exists with the evaluation

$$L_j(\tau )=\left(\mathcal{U}(F(\tau ))/a\right)j{\pi }_j{F}^{j-1}(\tau ).$$

(51)

However, the first factor equals $f(F(\tau ))-F(\tau )=1/{\pi }^{\prime }(F(\tau ))$, giving the value asserted for the first double limit. Moreover, ${\sum }_{j=1}^{\infty }{L}_j(\tau )=1$.

For the second double limit, observe that

$$E_i\left.\left(s^{Z_t}\right|t+\tau <T\le t+\tau +y\right)={\displaystyle \frac{F^i(sF(\tau +y),t)-F^i(sF(\tau ),t)}{F^i(t+\tau +y)-F^i(t+\tau )}}.$$

It follows from l’Hôpital’s rule that

$$\underset{y\to 0}{lim}{E}_i\left.\left(s^{Z_t}\right|t+\tau <T\le t+\tau +y\right)={\displaystyle \frac{s{\left.{(F(sF(\tau ),t))}^{i-1}\times F^{\prime }(\tau )\times \frac{\partial }{\partial s}F(s,t)\right|}_{s=sF(\tau )}}{{(F(t+\tau ))}^{i-1}\times F^{\prime }(t+\tau )}}.$$

We define this limit to be the pgf of $Z_t$ conditional on $T=t+\tau$ and denote it as $E_i\left.\left(s^{Z_t}\right|T=t+\tau \right)$.

Eliminating the time derivative in (48) using the analogous forward differential equation $\partial F/\partial t=\mathcal{U}(s)\partial F/\partial s$ yields

$${\displaystyle \frac{\partial F}{\partial s}}(s,t)={\displaystyle \frac{f(F(s,t))-F(s,t)}{f(s)-s}}$$

and, hence, the evaluation

$$E_i\left.\left(s^{Z_t}\right|T=t+\tau \right)={\left({\displaystyle \frac{F(sF(\tau ),t)}{F(t+\tau )}}\right)}^{i-1}·{\displaystyle \frac{s(f(F(\tau ))-F(\tau ))}{f(sF(\tau ))-sF(\tau )}}·{\displaystyle \frac{f(F(sF(\tau ),t)-F(sF(\tau ),t)}{f(F(t+\tau ))-F(t+\tau )}}.$$

Let $t\to \infty$ in the first quotient factor, observing that the numerator and denominator terms converge to unity.

For the third quotient factor, observe that for fixed numbers $s\in (0,1]$ and $\tau >0$, there exists a number $v_s>0$ such that $F(v_s)=sF(\tau )$. Since $F(t+\tau )=F(F(\tau ),t)$, it follows from (7.3) in [5] that

$$\underset{t\to \infty }{lim}{\displaystyle \frac{f(F(sF(\tau ),t)-F(sF(\tau ),t)}{f(F(t+\tau ))-F(t+\tau )}}=\underset{t\to \infty }{lim}{\displaystyle \frac{\mathcal{U}(F(t+v_s)}{\mathcal{U}(F(t+v_1)}}=1.$$

Hence,

$$\underset{t\to \infty }{lim}{E}_i\left.\left(s^{Z_t}\right|T=t+\tau \right)=s{\displaystyle \frac{f(F(\tau ))-F(\tau )}{f(sF(\tau ))-sF(\tau )}}=s{\displaystyle \frac{{\pi }^{\prime }(sF(\tau ))}{{\pi }^{\prime }(F(\tau ))}}$$

The right-hand side equals ${\sum }_{j=1}^{\infty }{L}_j(\tau )s^j$, and the proof is finished. □

It is clear that as $\tau$ ranges through $(0,\infty )$, the pgf

$$L(s,\tau ):=\sum _{j=1}^{\infty }{L}_j(\tau )s^j=s{\displaystyle \frac{{\pi }^{\prime }(sF(\tau ))}{{\pi }^{\prime }(F(\tau ))}}$$

(52)

specifies the power series family of laws derived from the weights $j{\pi }_j$. In addition, (51) can be expressed as

$$a{\pi }_j={\displaystyle \frac{L_j(\tau )}{(d/d\tau )F^j(\tau )}},\mathrm{i}.\mathrm{e}.,L_j(\tau )=a{\pi }_j{\displaystyle \frac{d}{d\tau }}{P}_j(T\le \tau ).$$

The right-hand side of the first equality is (up to scaling) the unique invariant measure of the transition semigroup, whose outcome is the MBP analogue of Theorem 2 in [9]. The second equality is the MBP analogue of (8).

6. Critical Case: Examples

A minimal desirable outcome for any example is obtaining explicit expressions for ${\pi }_j$ and/or ${\pi }^{\prime }(s)$, because this gives an identification of the limiting-power-series family whose pgf is $L(s,\tau )$. Better still is obtaining an explicit expression for $\pi (s)$ and its inverse $\mathcal{P}(y)$, because this would, from (49), yield the evaluation

$$F(s,t)=\mathcal{P}(\pi (s)+at).$$

(53)

We begin with two examples achieving the first aim.

Example 6. 

If $f(s)={(2-s)}^{-1}$, then

$$j{\pi }_j=\frac{1}{2}(j+1)\mathrm{and}F(\tau )=1-W\left(e^{1+a\tau }\right),$$

where $W(·)$ is the principal Lambert function. See [8] for the ramifications of this critical geometric offspring law and, in particular, evaluation of $\pi (s)$ and $\mathcal{P}(y)$. We return to this example as Example 10 below.

Example 7. 

If $f(s)=e^{s-1}$, the critical Poisson offspring law, then some algebra yields the evaluation

$$j{\pi }_j=e^{j-1}\sum _{i=0}^{j-1}{\displaystyle \frac{{(j-i)}^i}{i!}}{(-e^{-1})}^i,(j\ge 1).$$

It does not seem possible to evaluate $F(\tau )$ in an explicit form.

We now pursue a general scheme which yields examples of varying degrees of explicitness. Recalling that $g(s)=(1-f(s))/(1-s)$ is a pgf, it is observed in [5] (§7) that

$$U(s)=\sum _{j=0}^{\infty }{u}_j{s}^j={\displaystyle \frac{1}{1-g(s)}}$$

is the generating function of a renewal sequence, but a generalised one in the sense that, because $g_0=g(0)=1-p_0$, we have $U(0)=1/p_0>1$. Here, it is more convenient to define the modified pgf

$$\overline{g}(s)={\displaystyle \frac{g(s)-g_0}{1-g_0}}$$

which satisfies $\overline{g}(0)=0$, and hence,

$$V(s)=\sum _{j=0}^{\infty }{v}_j{s}^j={\displaystyle \frac{1}{1-\overline{g}(s)}}$$

is the generating function of a renewal sequence $(v_j:j\ge 0)$ which satisfies the standard normalisation $v_0=1$. However,

$$1-\overline{g}(s)=p_0^{-1}(1-g(s))=p_0^{-1}{\displaystyle \frac{f(s)-s}{1-s}}$$

and hence,

$$p_0{\pi }^{\prime }(s)=[(1-s){(1-\overline{g}(s)]}^{-1}={\displaystyle \frac{V(s)}{1-s}},$$

(54)

i.e.,

$$p_{0}j{\pi }_j=\sum _{i=0}^{j-1}{v}_i.$$

(55)

Example 8. 

Recall Example 6. We have $p_0=\frac{1}{2}$, so $p_0{\pi }_j=j+1$, and hence, from (55), if $j\ge 1$, then

$$v_j=\frac{1}{2}[(j+1){\pi }_{j+1}-j{\pi }_j]=\frac{1}{2}{\int }_0^1{u}^j\omega (du),$$

where $\omega (du)=\frac{1}{2}[{\delta }_0(du)+{\delta }_1(du)]$ is a probability measure on the closed unit interval $[0,1]$. So, in this example, $(v_j)$ is the moment sequence of a probability law on $[0,1]$.

This motivates the following strategy. Let $\omega (du)$ be an arbitrary probability measure on $[0,1]$. It is known that its moments

$$v_j={\int }_0^1{u}^j\omega (du)(j\ge 0)$$

(56)

comprise a renewal sequence, in fact, a Kaluza sequence, meaning that $v_j^2\le v_{j-1}{v}_{j+1}$. Evaluating the sum in (55) yields

$$p_{0}j{\pi }_j={\int }_0^1{\displaystyle \frac{1-u^j}{1-u}}\omega (du)$$

and hence,

$$p_0{\pi }^{\prime }(s)={(1-s)}^{-1}{\int }_0^1{\displaystyle \frac{\omega (du)}{1-us}},$$

i.e., compared with (54), we see that

$$V(s)={\int }_0^1{\displaystyle \frac{\omega (du)}{1-us}}.$$

(57)

Hence, starting from (56), we obtain the function

$$f(s)=s+p_0{\displaystyle \frac{1-s}{V(s)}}.$$

(58)

In order that $m=f^{\prime }(1)=1$, we require that

$$V(1)={\int }_0^1{\displaystyle \frac{\omega (du)}{1-u}}=\infty ,$$

(59)

a necessary condition. In addition,

$$p_1=f^{\prime }(0)=1-p_0(1+v_1)<1.$$

But we require that $p_1\ge 0$, i.e., a second necessary condition is that

$$p_0\le {\displaystyle \frac{1}{1+v_1}}.$$

(60)

Finally, since $V(s)$ is a renewal-generating function derived from a pgf, $\overline{g}(s)$ with ${\overline{g}}_0=0$, substituting it into (59) and expanding it yields

$$f(s)=p_0+\left(1-(1+v_1)p_0\right)s+p_0\sum _{j=2}^{\infty }({\overline{g}}_{j-1}-{\overline{g}}_j)s^j.$$

(61)

This yields the following result, which is the analogue of Theorems 9 and 10 above for the subcritical MBP.

Theorem 12. 

Suppose that $\omega (du)$ is a probability measure on $[0,1]$, and $f(s)$ is specified by (57) and (58). Then, $f(s)$ is a critical pgf iff (59) and (60) both hold and the sequence $({\overline{g}}_j:j\ge 1)$ is non-increasing.

Remark 7. 

It is obvious that the assertion will hold if $(v_j:j\ge 0)$ is a standard renewal sequence such that ${\sum }_{j=0}^{\infty }{v}_j=\infty$, the condition (60) holds, and the coefficients of $\overline{g}(s)=1-1/V(s)$ are non-decreasing.

Before looking at examples, observe that the integration of (57) yields

$$p_0\pi (s)={\int }_0^{1}log\left({\displaystyle \frac{1-us}{1-s}}\right){\displaystyle \frac{\omega (du)}{1-u}}$$

and that, despite (59), the integral is finite because the log-term equals

$${\displaystyle \frac{s}{1-s}}(1-u)(1+o(1))(u\uparrow 1).$$

It is easier to treat specific cases individually rather than using the above general expression.

Example 9. 

The simplest possible example for (56) is the uniform law on $[0,1]$. This yields $v_j={(1+j)}^{-1}$ and $V(s)=-s^{-1}log(1-s)$. Entry 140 in [22], in essence, states that

$$1-\overline{g}(s)={\int }_0^1{(1-s)}^{u}du=1-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{s^2}{12}}-{\displaystyle \frac{s^3}{24}}-\cdots .$$

More generally, expanding the integrand yields

$${\overline{g}}_j={\displaystyle \frac{1}{j!}}{\int }_0^{1}u{\displaystyle \frac{\mathsf{\Gamma }(j-u)}{\mathsf{\Gamma }(1-u)}}du(j\ge 1)$$

and

$${\overline{g}}_{j+1}<j{\overline{g}}_j/(j+1)<{\overline{g}}_j,$$

as required for Theorem 12. The condition (59) is satisfied and, since $v_1=\frac{1}{2}$, the condition (60) is satisfied if $0<p_0\le 2/3$.

It follows from (55) that $p_{0}j{\pi }_j=H(j)$, the j-th Harmonic number. Hence,

$$p_0\pi (s)=\sum _{j=1}^{\infty }{j}^{-1}H(j)s^j=s+\sum _{j=2}^{\infty }\left(H(j-1)+{\displaystyle \frac{1}{j}}\right){\displaystyle \frac{s^j}{j}}=\frac{1}{2}{\left(log(1-s)\right)}^2+{\mathrm{Li}}_2(s),$$

where we use Entry 111 in [22], and ${\mathrm{Li}}_2(s)$ is the dilogarithm function.

So, we obtain an explicit expression for $\pi (s)$ and its coefficients ${\pi }_j$, but the inverse $\mathcal{P}(y)$ looks elusive. The measure $\omega$ can be extended to $\omega (du)=p\nu u^{\nu -1}du+(1-p){\delta }_1(du)$, where $p\in (0,1]$ and $\nu >0$. If $\nu =1,2,\dots$, then

$$p_{0}j{\pi }_j=(1-p)j+p\nu [H(j+\nu -1)-H(\nu -1)],$$

where $H(0)=0$. Hence, $v_j=1-p+p\nu /(j+\nu )$, entailing a closed expression for $V(s)$. However, it does not seem possible to go further, even if $\nu =1$ or 2.

Example 10. 

Fix constants $\alpha ,p\in (0,1)$, $q=1-p$, and choose $\omega (du)=p{\delta }_{\alpha }(du)+q{\delta }_1(du)$, thus generalising Example 6. Hence, $v_j=q+p{\alpha }^j$,

$$V(s)={\displaystyle \frac{q}{1-s}}+{\displaystyle \frac{p}{1-\alpha s}},$$

whence $V(1)=\infty$. Defining $\beta =p+q\alpha \in (0,1)$ and noting that $1+\alpha -\beta =q+p\alpha \in (0,1)$, some algebra leads to

$$\overline{g}(s)=1-1/V(s)={\displaystyle \frac{(q+\alpha p)s-\alpha s^2}{1-\beta s}}=[q+\alpha p-\alpha s]\sum _{j=1}^{\infty }{\beta }^{j-1}{s}^j.$$

Hence, ${\overline{g}}_1=q+p\alpha$ and

$${\overline{g}}_j=[(q+\alpha p)\beta -\alpha ]{\beta }^{j-2}=pq{(1-\alpha )}^2{\beta }^{j-2},(j\ge 2).$$

Thus, ${\overline{g}}_j>{\overline{g}}_{j+1}$ if $j\ge 2$, as required. Referring to (58), we conclude that

$$f(s)=s+p_0{\displaystyle \frac{{(1-s)}^2(1-\alpha s)}{1-\beta s}}$$

is a critical pgf iff

$$0<p_0\le {(1+q+\alpha p)}^{-1}.$$

Continuing, we have

$$p_{0}j{\pi }_j=\sum _{i=0}^{j-1}{v}_i=qj+p{\displaystyle \frac{1-{\alpha }^j}{1-\alpha }}(j\ge 1)$$

and hence,

$$p_0\pi (s)={\displaystyle \frac{qs}{1-s}}-{\displaystyle \frac{p}{1-\alpha }}log(1-s)+{\displaystyle \frac{p}{1-\alpha }}log(1-\alpha s).$$

Inverting this expression seems intractable except if $\alpha =0$, i.e., $\beta =p$.

In this case,

$$p_0\pi (s)={\displaystyle \frac{qs}{1-s}}-plog(1-s)$$

(62)

and

$$f(s)=s+p_0{\displaystyle \frac{{(1-s)}^2}{1-ps}}$$

(63)

is a critical pgf iff

$$0<p_0\le {(1+q)}^{-1}.$$

Writing $\zeta ={(1-s)}^{-1}$, $y=\pi (s)$ and $r=q/p$, the Equation (62) can be recast as $r\zeta +log\zeta =x:=r(1+p_{0}y/q)$, i.e., $r\zeta e^{r\zeta }=r{e}^x$. This is the Lambert functional equation, whose solution is $\zeta =r^{-1}W(r{e}^x)$, where, because $\zeta >1$, $W(·)$ is the principal Lambert function. It follows that the inverse of $\pi (s)$ is

$$\mathcal{P}(y)=1-r/W\left(r{e}^{r(1+p_{0}y/q)}\right).$$

Thus, an explicit, though complicated, expression for $F(s,t)$ follows from (53) and (62). In particular, we have a complete identification of the limiting conditional law specified by (52):

$$j{\pi }_j=(p+qj)/p_0\mathrm{and}F(\tau )=\mathcal{P}(a\tau ).$$

Example 6 is recovered when $p_0=p=\frac{1}{2}$.

We can say a little more about this case by observing that (63) can be expressed as

$$f(s)=1-{\displaystyle \frac{(1-s)(1-p_0+(p_0-p)s)}{1-ps}}.$$

Setting $p_0=p=C/(1+C)$, where $C>0$, this becomes

$$f(s)=1-{\left({(1+s)}^{-1}+C\right)}^{-1}.$$

This family of critical offspring pgfs coincides with the critical theta-offspring laws which have a finite variance identified and explored as Case 2 in [23].

A more direct way of achieving specifications of $\pi (s)$ and $f(s)$ is simply to select a mass function $({\overline{g}}_j)$ with decreasing masses. We then have

$$f(s)=1-(1-s)[1-p_0+p_0\overline{g}(s)]=p_0[1-p_0(1+{\overline{g}}_1)]s+p_0\sum _{j=2}^{\infty }({\overline{g}}_{j-1}-{\overline{g}}_j)s^j$$

and this is a critical offspring pgf iff

$$0<p_0\le {(1+{\overline{g}}_1)}^{-1}.$$

(64)

Example 11. 

Set $\overline{g}(s)=s{[(1-p)/(1-ps)]}^{\sigma }$, a shifted negative-binomial law. The decreasing property holds iff $\sigma p\le 1$ and we have ${\overline{g}}_1={(1-p)}^{\sigma }$. It does not seem possible to go much further unless σ assumes small integer values. The case $\sigma =1$ is the particular instance of Example 10 with $\alpha =0$.

If $\sigma =2$, then we require $p\le \frac{1}{2}$. Defining $A=(1-p)/(1+p)$, some algebra yields

$$V(s)={\displaystyle \frac{1-2ps+p^2{s}^2}{1-(1+p^2)s+p^2{s}^2}}=1+A\left({\displaystyle \frac{1}{1-s}}-{\displaystyle \frac{1}{1-p^{2}s}}\right)={\int }_0^1{\displaystyle \frac{\omega (du)}{1-su}},$$

where

$$\omega (du)={\delta }_0(du)-A{\delta }_{p^2}(du)+A{\delta }_1(du)$$

is a signed measure with unit total mass. Hence,

$$v_0=1\mathrm{and}{v}_j=A(1-p^{2j}),(j\ge 1).$$

It follows that the stationary measure is given by ${\pi }_1=1/p_0$, and

$$p_{0}j{\pi }_j=1+A\left(j-1-{\displaystyle \frac{1-p^{2j}}{1-p^2}}\right),(j\ge 2).$$

Example 12. 

Another simple choice is ${\overline{g}}_j=1/j(j+1)$, in which case we require that $0<p_0\le 2/3$. Also,

$$\overline{g}(s)=1+\left(s^{-1}-1\right)log(1-s)$$

and

$$V(s)={\displaystyle \frac{s}{(1-s)log\frac{1}{1-s}}}.$$

Hence,

$$p_1=1-3p_0/2\mathrm{and}{p}_j={\displaystyle \frac{2p_0}{j(j^2-1)}},(j\ge 2).$$

Let $\sigma \ge 0$ and consider the function

$$V_{\sigma }(s)={\displaystyle \frac{s}{{(1-s)}^{\sigma }log\frac{1}{1-s}}}={\int }_0^1{(1-s)}^{u-\sigma }du.$$

The integrand can be expanded as a power series and the result integrated term-wise. If $\sigma \ge 1$, this yields

$$V_{\sigma }(s)=\sum _{j=0}^{\infty }{v}_j(\sigma )s^j\mathrm{where}{v}_j(\sigma )={\displaystyle \frac{1}{j!}}{\int }_0^1{\displaystyle \frac{\mathsf{\Gamma }(j+\sigma -u)}{\mathsf{\Gamma }(\sigma -u)}}du.$$

In particular, $v_j=v_j(1)$ and $p_0{\pi }^{\prime }(s)=V_2(s)$, implying that

$$p_0{\pi }_j={\displaystyle \frac{1}{j!}}{\int }_0^1{\displaystyle \frac{\mathsf{\Gamma }(j+1-u)}{\mathsf{\Gamma }(2-u)}}du.$$