Single-Block Recursive Poisson–Dirichlet Fragmentations of Normalized Generalized Gamma Processes

Abstract

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for $0<\alpha <1,$ and $\theta >-\alpha$, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters $(\alpha ,1-\alpha )$ to a mass partition having a Poisson–Dirichlet distribution with parameters $(\alpha ,\theta )$ leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters $(\alpha ,\theta +r)$ for $r=0,1,2,\cdots .$ Furthermore, these generate a Markovian sequence of $\alpha$-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer $n,$ which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where $n=0,1.$

1. Introduction

Let $\mathbf{Z}=(Z_r,r\ge 0)$ denote a Markov chain characterized by a stationary transition density $Z_r|Z_{r-1}=z$ given for $y>z$ and $0<\alpha <1$:

$$\mathbb{P}(Z_r\in dy|Z_{r-1}=z)/dy=\frac{\alpha {(y-z)}^{\frac{1-\alpha }{\alpha }-1}y{g}_{\alpha }\left(y\right)}{\Gamma \left(\frac{1-\alpha }{\alpha }\right)g_{\alpha }\left(z\right)},$$

(1)

where $g_{\alpha }\left(s\right):=f_{\alpha }\left(s^{-\frac{1}{\alpha }}\right)s^{-\frac{1}{\alpha }-1}/\alpha$ is the density of a variable $T_{\alpha }^{-\alpha },$ with a Mittag-Leffler distribution, $T_{\alpha }:=T_{\alpha ,0}$ is a positive stable variable with density denoted as $f_{\alpha }\left(t\right),$ and Laplace transform $\mathbb{E}\left[e^{-\lambda T_{\alpha }}\right]=e^{-{\lambda }^{\alpha }}.$ More generally, as in [1,2,3,4], for $\theta >-\alpha ,$ let $T_{\alpha ,\theta }$ denote a variable with density $f_{\alpha ,\theta }\left(t\right)=t^{-\theta }{f}_{\alpha }\left(t\right)/\mathbb{E}\left[T_{\alpha }^{-\theta }\right];$ then, $T_{\alpha ,\theta }^{-\alpha }$ is said to have a generalized Mittag-Leffler distribution with parameters $(\alpha ,\theta )$ and distribution denoted as $\mathrm{ML}(\alpha ,\theta ).$ In the cases where $Z_0=T_{\alpha ,\theta }^{-\alpha }\sim \mathrm{ML}(\alpha ,\theta ),$ the marginal distributions of each $Z_r$ are $\mathrm{ML}(\alpha ,\theta +r).$ Furthermore, there is a sequence of random variables $(B_j,j\ge 1)$ defined for each integer j as $B_j=Z_{j-1}/Z_j$; hence, there is the exact point-wise relation $Z_{j-1}=Z_j\times B_j,$ where, remarkably, the $B_j$ are independent $\mathrm{Beta}(\frac{\theta +\alpha +j-1}{\alpha },\frac{1-\alpha }{\alpha })$ variables, and $(B_1,\dots ,B_j)$ is independent of $Z_j,$ for $j=1,2,\dots .$ Note further that by setting $Z_r=T_{\alpha ,\theta +r}^{-\alpha },$ there is the point-wise equality $T_{\alpha ,\theta }=T_{\alpha ,\theta +r}\times {\prod }_{j=1}^r{B}_j^{-\frac{1}{\alpha }},$ where all the variables on the right-hand side are independent. In these cases, the sequence may be referred to as a Mittag-Leffler Markov chain with law denoted as $\mathbf{Z}\sim \mathrm{MLMC}(\alpha ,\theta ),$ as in [5] and, subsequently, [6]. The Markov chain is described prominently in various generalities, that is, ranges of $\alpha$ and $\theta$, in [5,6,7,8,9]. See for example [5,6,10,11,12,13,14,15] for more references concerning Pólya urn and random tree/graph growth models.

Now, let $\mathrm{PD}(\alpha ,\theta )$ denote a two-parameter Poisson–Dirichlet distribution over the space of mass partitions summing to $1,$ say ${\mathcal{P}}_{\infty }:=\{\mathbf{s}=(s_1,s_2,\dots ):s_1\ge s_2\ge \cdots \ge 0$ $\mathrm{and}{\sum }_{i=1}^{\infty }{s}_i=1\}$, as described in [3,4,16]. Let $\left(P_{\ell }\right):=(\left(P_{\ell }\right),\ell \ge 1)\sim \mathrm{PD}(\alpha ,\theta )$ correspond in distribution to the ranked lengths of excursion of a generalized Bessel bridge on $[0,1],$ as described and defined in [1,4]. In particular, $\mathrm{PD}(1/2,0)$ and $\mathrm{PD}(1/2,1/2)$ correspond to excursion lengths of standard Brownian motion and Brownian bridge, on $[0,1],$ respectively. As noted in [6], the single-block $\mathrm{PD}(\alpha ,1-\alpha )$ fragmentation results for $\mathrm{PD}(\alpha ,\theta )$ mass partitions by [17], which we shall describe in more detail in Section 1.2, allow one to couple a version of $\mathbf{Z}\sim \mathrm{MLMC}(\alpha ,\theta )$ with a nested family of mass partitions $(\left(P_{\ell ,r}\right),r\ge 0),$ where each $\left(P_{\ell ,r}\right):=(\left(P_{\ell ,r}\right),\ell \ge 1)$ takes its values in ${\mathcal{P}}_{\infty }$, initial $\left(P_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,\theta )$ has $\alpha$-diversity $Z_0=T_{\alpha ,\theta }^{-\alpha }$, and each successive $\left(P_{\ell ,r}\right)\sim \mathrm{PD}(\alpha ,\theta +r)$ has $\alpha$-diversity $Z_r=T_{\alpha ,\theta +r}^{-\alpha }.$ The distribution of this family is denoted as $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta ).$

Recall from [2] that for $\left(P_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,0),$$\left(P_{\ell ,0}\right)|T_{\alpha }=t$ has distribution $\mathrm{PD}\left(\alpha \right|t)$, and for a probability measure $\nu$ on $(0,\infty )$, one may generate the general class of Poisson–Kingman distributions generated by an $\alpha$-stable subordinator with mixing $\nu ,$ by forming ${\mathrm{PK}}_{\alpha }\left(\nu \right)={\int }_0^{\infty }\mathrm{PD}\left(\alpha \right|t)\nu \left(dt\right).$ Some prominent examples of interest in this work are $\mathrm{PD}(\alpha ,\theta )={\int }_0^{\infty }\mathrm{PD}\left(\alpha \right|t)f_{\alpha ,\theta }\left(t\right)dt$ and ${\mathbb{P}}_{\alpha }^{\left[n\right]}\left(\lambda \right)={\int }_0^{\infty }\mathrm{PD}\left(\alpha \right|t)f_{\alpha }^{\left[n\right]}\left(t\right|\lambda )dt,$ where $f_{\alpha }^{\left[n\right]}\left(t\right|\lambda )\propto t^n{e}^{-\lambda t}{f}_{\alpha }\left(t\right).$ Hence, ${\mathbb{P}}_{\alpha }^{\left[0\right]}\left(\lambda \right)$ corresponds to the law of the ranked normalized jumps of a generalized gamma subordinator, say $({\tau }_{\alpha }\left(y\right);y\ge 0),$ where ${\tau }_{\alpha }\left({\lambda }^{\alpha }\right)/\lambda$ has density $f_{\alpha }^{\left[0\right]}\left(t\right|\lambda )=e^{-\lambda t}{e}^{{\lambda }^{\alpha }}{f}_{\alpha }\left(t\right).$ In [6], we obtained some general distributional properties of $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)$ formed by repeated application of the fragmentation operations in [17] to the case where $\left(P_{\ell ,0}\right)\sim {\mathrm{PK}}_{\alpha }\left(\nu \right).$ Furthermore, letting $\left({\mathbf{e}}_{\ell }\right)$ denote a sequence of iid $\mathrm{Exp}\left(1\right)$ variables forming the arrival times, say $({\Gamma }_{\ell }={\sum }_{j=1}^{\ell }{\mathbf{e}}_j;\ell \ge 1)$, of a standard Poisson process, we ([6], Section 4.3) focused in more detail on the special case of $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)|N_{T_{\alpha ,\theta }^{-\alpha }}\left(\lambda \right)=j$ for $j=0,1,2,\dots ,$ when $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta )$ and $(N_{T_{\alpha ,\theta }^{-\alpha }}\left(t\right)={\sum }_{\ell =1}^{\infty }{\mathbb{I}}_{\{{\Gamma }_{\ell }/T_{\alpha ,\theta }^{-\alpha }\le t\}},t\ge 0)$ is a mixed Poisson process with random intensity depending on $T_{\alpha ,\theta }^{-\alpha }.$ That is to say, $\left(P_{\ell ,0}\right)|N_{T_{\alpha ,\theta }^{-\alpha }}\left(\lambda \right)=j$ corresponds in distribution to $\left(P_{\ell ,0}\left(\lambda \right)\right)$ following a ${\mathrm{PK}}_{\alpha }\left(\nu \right)$ distribution, where $\nu$ corresponds to the distribution of $T_{\alpha ,\theta }^{-\alpha }|N_{T_{\alpha ,\theta }^{-\alpha }}\left(\lambda \right)=j.$

In this work, we obtain results for the case where $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)$ is such that $\left(P_{\ell ,0}\right)\sim {\mathbb{P}}_{\alpha }^{\left[n\right]}\left(\lambda \right),$ which is when $\left(P_{\ell ,0}\right)$ corresponds to the ranked normallized jumps of a generalized gamma process, $({\tau }_{\alpha }\left(y\right);y\ge 0),$ and its size-biased generalizations. Interestingly, our results equate in distribution to the following setup involving $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0).$ Let $N_{T_{\alpha }}$ be a mixed Poisson process defined by replacing $T_{\alpha ,\theta }^{-\alpha }$ in $N_{T_{\alpha ,\theta }^{-\alpha }}$ with $T_{\alpha }.$ Using the mixed Poisson framework in the manuscript of Pitman [18] (see also [6,19] for more details), we obtain some explicit distributional properties of $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)|N_{T_{\alpha }}\left(\lambda \right)=n$ and corresponding variables $(B_1,\dots ,B_r,T_{\alpha ,r})|N_{T_{\alpha }}\left(\lambda \right)=n$ for $n=0,1,2,\dots ,$ when $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0).$ That is when $\left(P_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,0).$ The equivalence in distribution to the fragmentation operations of [17] applied in the generalized gamma cases may be deduced from [18], who shows that when $\left(P_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,0),$$\left(P_{\ell ,0}\right)|N_{T_{\alpha }}=n$ corresponds to the distribution of $\left(P_{\ell ,0}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[n\right]}\left(\lambda \right).$ We shall primarily focus on the case of $n=0,1,$ corresponding to the generalized gamma density and its sized biased distribution, which yields the most explicit results. The fragmentation operations (6) applied to $\left(\left(P_{\ell ,0}\right)\right)\sim P_{\alpha }^{\left[1\right]}\left(\lambda \right)$ allow one to recover the entire range of $\mathrm{PD}(\alpha ,\theta )$ distributions for $\theta >-\alpha ,$ by gamma randomization, whereas the case for $\left(\left(P_{\ell ,0}\right)\right)\sim P_{\alpha }^{\left[0\right]}\left(\lambda \right)$ only applies to $\theta \ge 0.$ We note that descriptions of our results for $n=0,1$, albeit less refined ones, appear in the unpublished manuscript ([9], Section 6). See also [20] for an application of $P_{\alpha }^{\left[0\right]}\left(\lambda \right)$ for randomized $\lambda .$

We close this section by recalling the definition of the first size-biased pick from a random mass partition $\left(P_{\ell }\right)\in {\mathcal{P}}_{\infty }$ (see [2,3,16]). Specifically, ${\tilde{P}}_1$ is referred to as the first size-biased pick from $\left(P_{\ell }\right),$ if it satisfies, for $k=1,2,\dots ,$

$$\mathbb{P}({\tilde{P}}_1=P_k|\left(P_{\ell }\right))=P_k.$$

(2)

Hereafter, let ${\left(P_{\ell }\right)}_1:=\left(P_{\ell }\right)\backslash {\tilde{P}}_1$ denote the remainder, such that $\left(P_{\ell }\right)=\mathrm{Rank}\left(({\left(P_{\ell }\right)}_1,{\tilde{P}}_1)\right),$ where $\mathrm{Rank}(·)$ denotes the operation corresponding to ranked re-arrangement. From [1], ${\tilde{P}}_1$ may be interpreted as the length of excursion (i.e., one of the $\left(P_{\ell }\right)$), first discovered by dropping a uniformly distributed random variable onto the interval $[0,1].$ The fragmentation operation of [17] may be interpreted as shattering/fragmenting that interval by the excursion lengths of a process on $[0,1],$ with distribution $\mathrm{PD}(\alpha ,1-\alpha )$ and then re-ranking. For clarity and comparison, we first recall some details of the more well-known Markovian size-biased deletion operation leading to stick-breaking representations, as described in [1,2,3], and more related notions arising in a Bayesian nonparametric context in the $\mathrm{PD}(\alpha ,\theta )$ setting, in the next section.

Remark 1.

Although we acknowledge the influence and contributions of the manuscript [18], the pertinent distributional results we use from that work are re-derived at the beginning of Section 2. Otherwise, the interpretation of $N_{T_{\alpha }}$ from that work is briefly mentioned in Section 1.3.

1.1. $PD(\alpha ,\theta )$ Markovian Sequences Obtained from Successive Size-Biased Deletion

Following [1], we may define $\mathrm{SBD}(·)$ to be a size-biased deletion operator on ${\mathcal{P}}_{\infty },$ as $\mathrm{SBD}\left(\left(P_{\ell }\right)\right):=\mathrm{Rank}\left(({\left(P_{\ell }\right)}_1/(1-{\tilde{P}}_1))\right),$ where it can be recalled from (2) that $\left(P_{\ell }\right)=\mathrm{Rank}\left(({\left(P_{\ell }\right)}_1,{\tilde{P}}_1)\right).$ Now, let $({\mathrm{SBD}}^{\left(j\right)}(·),j\ge 1)$ be a collection of such operators. From [1], as per the description in ([4], Proposition 34, p. 881), it follows that for $\left(P_{\ell ,0}\right):=\left({\widehat{P}}_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,\theta ),$${\mathrm{SBD}}^{\left(1\right)}\left(\left({\widehat{P}}_{\ell ,0}\right)\right):=\left({\widehat{P}}_{\ell ,1}\right)\sim \mathrm{PD}(\alpha ,\theta +\alpha )$ and is independent of the first size-biased pick ${\tilde{P}}_1:=V_1\sim \mathrm{Beta}(1-\alpha ,\theta +\alpha )$, and hence, for $r=2,\dots ,$

$$\left({\widehat{P}}_{\ell ,r}\right):={\mathrm{SBD}}^{\left(r\right)}\left(\left({\widehat{P}}_{\ell ,r-1}\right)\right)={\mathrm{SBD}}^{\left(r\right)}\circ \cdots \circ {\mathrm{SBD}}^{\left(1\right)}\left(\left({\widehat{P}}_{\ell ,0}\right)\right)\sim \mathrm{PD}(\alpha ,\theta +r\alpha ).$$

(3)

This leads to a nested Markovian family of mass partitions $(\left({\widehat{P}}_{\ell ,r}\right),r\ge 0),$ where $\left(P_{\ell ,0}\right):=\left({\widehat{P}}_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,\theta )$ with inverse local time at time $1,$ $T_{\alpha ,\theta }$(see ([3], Equation (4.20), p. 83)), and for each $r,$$\left({\widehat{P}}_{\ell ,r}\right)\sim \mathrm{PD}(\alpha ,\theta +r\alpha )$ with inverse local time at time $1,$ $T_{\alpha ,\theta +r\alpha }.$ Furthermore, $(T_{\alpha ,\theta +r\alpha },r\ge 0)$ form a Markov chain with pointwise equality $T_{\alpha ,\theta +(j-1)\alpha }=T_{\alpha ,\theta +j\alpha }/(1-V_j),$ where $V_j$ are independent $\mathrm{Beta}(1-\alpha ,\theta +j\alpha )$ variables and are the respective first size-biased picks from $\left({\widehat{P}}_{\ell ,j-1}\right)$ for $j\ge 1.$ Furthermore, $(V_1,\dots ,V_r)$ is independent of $T_{\alpha ,\theta +r\alpha }$ and, more generally, $\left({\widehat{P}}_{\ell ,r}\right)$ for $r=1,2,\dots .$

From this, one obtains the size-biased re-arrangement of a $\mathrm{PD}(\alpha ,\theta )$ mass partition, say $\left({\tilde{P}}_{\ell }\right)\sim \mathrm{GEM}(\alpha ,\theta ),$ satisfying ${\tilde{P}}_1=V_1\sim \mathrm{Beta}(1-\alpha ,\theta +\alpha )$, and for $\ell \ge 2,$${\tilde{P}}_{\ell }=V_{\ell }{\prod }_{j=1}^{\ell -1}(1-V_j).$ Refs. [3,21] discuss the $\mathrm{GEM}(\alpha ,\theta )$ distribution and these other concepts in a species sampling and Bayesian context. We mention the roles of corresponding random distribution functions as priors in a Bayesian non-parametric context. Let $\left(U_{\ell }\right)$ denote a sequence of iid $\mathrm{Uniform}[0,1]$ variables independent of $\left(P_{\ell }\right)\sim \mathrm{PD}(\alpha ,\theta );$ then, the random distribution $F_{\alpha ,\theta }\left(y\right)={\sum }_{\ell =1}^{\infty }{P}_{\ell }{\mathbb{I}}_{\{U_{\ell }\le y\}}$ is said to follow a Pitman–Yor distribution with parameters $(\alpha ,\theta ),$ (see [21,22]). $F_{\alpha ,\theta }$ is a two-parameter extension of the Dirichlet process [23] (which corresponds to $F_{0,\theta }$) and has been applied extensively as a more flexible prior in a Bayesian context, but it also arises in a variety of areas involving combinatorial stochastic processes [3,21]. An attractive feature of $F_{\alpha ,\theta }$ is that it may be represented as $F_{\alpha ,\theta }\left(y\right)={\sum }_{\ell =1}^{\infty }{\tilde{P}}_{\ell }{\mathbb{I}}_{\{{\tilde{U}}_{\ell }\le y\}},$ where $\left({\tilde{U}}_{\ell }\right)$ are the iid $\mathrm{Uniform}[0,1]$ concomittants of the $\left({\tilde{P}}_{\ell }\right),$ as exploited in [22] (see also [21]). This constitutes the stick-breaking representation of $F_{\alpha ,\theta }.$ Furthermore, we can describe ${\tilde{P}}_1$ as folllows: let $X_1|F_{\alpha ,\theta }$ have distribution $F_{\alpha ,\theta },$ and denote the first value drawn from $F_{\alpha ,\theta }$; then, ${\tilde{P}}_1$ is the mass in $\left(P_{\ell }\right)$ corresponding to that atom of $F_{\alpha ,\theta }.$ The size-biased deletion operation described above, as in (3), leads to the following decomposition of $F_{\alpha ,\theta }:$

$$F_{\alpha ,\theta }\left(y\right)=(1-{\tilde{P}}_1)F_{\alpha ,\theta +\alpha }\left(y\right)+{\tilde{P}}_1{\mathbb{I}}_{\{{\tilde{U}}_1\le y\}}$$

(4)

where $({\tilde{P}}_1,{\tilde{U}}_1)$ are independent of $F_{\alpha ,\theta +\alpha }\left(y\right)\stackrel{d}{=}{\sum }_{k=1}^{\infty }{\widehat{P}}_{k,1}{\mathbb{I}}_{\{U_{k,1}\le y\}},$ where $\left({\widehat{P}}_{\ell ,1}\right)\sim \mathrm{PD}(\alpha ,\theta +\alpha )$, and independent of this, where $\left(U_{\ell ,1}\right)\stackrel{iid}{\sim }\mathrm{Uniform}[0,1].$ See [1,4,24] and references therein for various interpretations of (4).

1.2. DGM Fragmentation

The single-block $\mathrm{PD}(\alpha ,1-\alpha )$ fragmentation operator of [17] is defined over the space ${\mathcal{P}}_{\infty }.$ However, for further clarity we start with an explanation at the level of random distribution functions involving the representation in (4). Suppose that $G_{\alpha ,1-\alpha }\left(y\right):={\sum }_{k=1}^{\infty }{Q}_k{\mathbb{I}}_{\{U_{k,1}^{\prime }\le y\}}$, with $\left(Q_{\ell }\right)\sim \mathrm{PD}(\alpha ,1-\alpha )$ and, independent of this, $\left(U_{\ell ,1}^{\prime }\right)\stackrel{iid}{\sim }\mathrm{Uniform}[0,1];$ hence, $G_{\alpha ,1-\alpha }\stackrel{d}{=}{F}_{\alpha ,1-\alpha }.$ Suppose that $G_{\alpha ,1-\alpha }$ is chosen independent of $F_{\alpha ,\theta }$ in (4); then, it follows from [17] that

$$F_{\alpha ,\theta +1}\left(y\right)\stackrel{d}{=}(1-{\tilde{P}}_1)F_{\alpha ,\theta +\alpha }\left(y\right)+{\tilde{P}}_1{G}_{\alpha ,1-\alpha }\left(y\right),$$

(5)

and it is evident that the mass partition $\left(Q_{\ell }\right)$ shatters/fragments ${\tilde{P}}_1$ into a countably infinite number of pieces $\left({\tilde{P}}_1\left(Q_{\ell }\right)\right):=({\tilde{P}}_1{Q}_{\ell },\ell \ge 1)=({\tilde{P}}_1{Q}_1,{\tilde{P}}_1{Q}_2,\dots ).$ It follows that, in this case, $\mathrm{Rank}\left({\left(P_{\ell }\right)}_1,{\tilde{P}}_1\left(Q_{\ell }\right)\right)\sim \mathrm{PD}(\alpha ,\theta +1),$ which is the featured case of the $\mathrm{PD}(\alpha ,1-\alpha )$ fragmentation described in [17]. Hence, for general $\left(P_{\ell }\right)=\mathrm{Rank}\left(({\left(P_{\ell }\right)}_1,{\tilde{P}}_1)\right)\in {\mathcal{P}}_{\infty },$ a $\mathrm{PD}(\alpha ,1-\alpha )$ fragmentation of $\left(P_{\ell }\right)$ is defined as

$${\widehat{\mathrm{Frag}}}_{\alpha ,1-\alpha }\left(\left(P_{\ell }\right)\right):=\mathrm{Rank}\left(({\left(P_{\ell }\right)}_1,{\tilde{P}}_1\left(Q_{\ell }\right))\right)\in {\mathcal{P}}_{\infty },$$

where, independent of $\left(P_{\ell }\right),$$\left(Q_{\ell }\right)\sim \mathrm{PD}(\alpha ,1-\alpha ).$ Let $\left(\left(Q_{\ell }^{\left(j\right)}\right);j\ge 1\right)$ denote an independent collection of $\mathrm{PD}(\alpha ,1-\alpha )$ mass partitions defining a sequence of independent fragmentation operators $\left({\widehat{\mathrm{Frag}}}_{\alpha ,1-\alpha }^{\left(j\right)}(·);j\ge 1\right).$ It follows from [17] that a version of the family $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta )$ may be constructed by the recursive fragmentation, for $r=1,2,\dots :$

$$\left(P_{\ell ,r}\right)={\widehat{\mathrm{Frag}}}_{\alpha ,1-\alpha }^{\left(r\right)}\left(\left(P_{\ell ,r-1}\right)\right)$$

(6)

In particular, $\left(P_{\ell ,r}\right)\sim \mathrm{PD}(\alpha ,\theta +r)$ when $\left(P_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,\theta ).$

1.3. Remarks

We close this section with remarks related to some relevant work of Eugenio Regazzini and his students, arising in a Bayesian context. From [18], in regards to a species sampling context using $F_{\alpha ,\theta }$ (see [21]), $N_{T_{\alpha ,\theta }}\left(\lambda \right)$ interprets as the number of animals trapped and tagged up until time $\lambda ,$ and hence, ${\Gamma }_j/T_{\alpha ,\theta }$ interprets as the time when the j-th animal is trapped for $j=1,\dots .$ Ref. [18] indicates that this gives further interpretation to such types of quantities arising in [25,26]. Using a Chinese restaurant process metaphor, the animals may be replaced by customers arriving sequentially to a restaurant. More generically, $N_{T_{\alpha ,\theta }}\left(\lambda \right)$ is the number of exchangeable samples drawn from $F_{\alpha ,\theta }$ up until time $\lambda .$ Furthermore, $F_{\alpha ,n}\left(y\right)|N_{T_{\alpha ,n}}\left(\lambda \right)=n$ for each $n=0,1,2,\dots$ is equivalent in distribution to $F_{\alpha }\left(y\right|\lambda )\stackrel{d}{=}{\tau }_{\alpha }\left({\lambda }^{\alpha }y\right)/{\tau }_{\alpha }\left({\lambda }^{\alpha }\right),$ which is now referred to in the Bayesian literature as a normalized generalized gamma process. While, according to [2], $F_{\alpha }\left(y\right|\lambda )$ appears in a relevant species sampling context in the 1965 thesis of McCloskey [27], and certainly elsewhere, the paper by Reggazzini, Lijoi, and Prünster [28] and subsequent works by Regazzini’s students (see [29]) helped to popularize the usage of $F_{\alpha }\left(y\right|\lambda )$ in the modern literature on Bayesian non-parametrics. Our work presents a view of $F_{\alpha }\left(y\right|\lambda )$ subjected to the fragmentation operations in [17]. Although we do not consider specific Bayesian statistical applications in this work, we note that other types of fragmentation/coagulation of $\mathrm{PD}(\alpha ,\theta )$ models have been applied, for instance, in [30]. We anticipate the same will be true of the operations considered here.

2. Results

Hereafter, we shall focus on the case of $\mathrm{PD}(\alpha ,0),$ as we will recover the general $(\alpha ,\theta )$ cases by applying gamma randomization as in ([4], Proposition 21) for $\theta \ge 0$ or ([19], Corollary 2.1) for $\theta >-\alpha$ and other results. See also ([6], Section 2.2.1). We first re-derive some relevant properties related to $N_{T_{\alpha }}$ that are easily verified by first conditioning on $T_{\alpha }$ and otherwise can be found in [18]. First, for fixed $\lambda ,$ and for $j=0,1,\dots ,$

$$\mathbb{P}(N_{T_{\alpha }}\left(\lambda \right)=j,T_{\alpha }\in ds)=\frac{{\lambda }^j}{j!}{s}^j{e}^{-\lambda s}{f}_{\alpha }\left(s\right)ds,$$

(7)

and for $j=1,2,\dots ,$

$$\mathbb{P}\left(\frac{{\Gamma }_j}{T_{\alpha }}\in d\lambda ,T_{\alpha }\in ds\right)/d\lambda =\frac{{\lambda }^{j-1}}{(j-1)!}{s}^j{e}^{-\lambda s}{f}_{\alpha }\left(s\right)ds.$$

(8)

Note these simple results hold for any variable T with density $f_T$ in place of $T_{\alpha }$ and $f_{\alpha }.$ It follows from (7) and (8) that $T_{\alpha }|N_{T_{\alpha }}\left(\lambda \right)=0$ has the generalized gamma density $f_{\alpha }^{\left[0\right]}\left(t\right|\lambda )=e^{-\lambda t}{e}^{{\lambda }^{\alpha }}{f}_{\alpha }\left(t\right).$ Furthermore, for $j=1,2,\dots ;$$T_{\alpha }|N_{T_{\alpha }}\left(\lambda \right)=j$ has the same distribution as $T_{\alpha }|{\Gamma }_j/T_{\alpha }=\lambda$ with density $f_{\alpha }^{\left[j\right]}\left(t\right|\lambda ).$ Since it is assumed that $({\Gamma }_{\ell };\ell \ge 1)$ is independent of $\left(P_{\ell }\right),$ it follows that for $\left(P_{\ell }\right)\sim \mathrm{PD}(\alpha ,0),$ the conditional distribution of $\left(P_{\ell }\right)|T_{\alpha }=t,N_{T_{\alpha }}\left(\lambda \right)=n$ is $\mathrm{PD}\left(\alpha \right|t)$, and hence, $\left(P_{\ell }\right)|N_{T_{\alpha }}\left(\lambda \right)=n$ has distribution ${\mathbb{P}}_{\alpha }^{\left[n\right]}\left(\lambda \right)$ for $n=0,1,\dots ,$ as mentioned previously.

Remark 2.

For the next results, which are extensions to $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0),$ conditioned on $N_{T_{\alpha }}\left(\lambda \right)=n,$ we note, as in [19], that the densities $f_{\alpha }^{\left[n\right]}\left(t\right|\lambda )$ are well-defined for any real number ϱ in place of $\left[n\right],$ with density $f_{\alpha }^{\left[\varrho \right]}\left(t\right|\lambda ),$ provided that $\lambda >0,$ and for $\lambda =0$ only in the case where $\varrho =-\theta <\alpha ,$ which corresponds to $f_{\alpha ,\theta }\left(t\right).$ Ref. ([19], Corollary 2.1) shows that distributions for ϱ can be expressed as randomized (over λ) distributions for any $n>\varrho .$

For clarity, with respect to $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0),$ $B_j=Z_{j-1}/Z_j$ are independent $\mathrm{Beta}(\frac{\alpha +j-1}{\alpha },\frac{1-\alpha }{\alpha })$ variables for $j=1,2,\dots ,$ and $(B_1,\dots ,B_r)$ is independent of $Z_r=T_{\alpha ,r}^{-\alpha }$ and $\left(P_{\ell ,r}\right)$ for each $r=1,2,\dots .$

Proposition 1.

Consider $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0),$ formed by the fragmentation operations in (6), when $\left(P_{\ell ,0}\right)\sim \mathrm{PD}(\alpha ,0).$ Denote the conditional distribution of $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)|N_{T_{\alpha }}\left(\lambda \right)=n$ as ${\mathrm{MLMC}}_{\mathrm{frag}}^{\left[n\right]}\left(\alpha \right|\lambda )$ and its corresponding component values as $(\left(P_{\ell ,r}\left(\lambda \right)\right),Z_r\left(\lambda \right);r\ge 0).$ Then, the distribution has the following properties.

(i) 

$\left(P_{\ell ,0}\right)|N_{T_{\alpha }}\left(\lambda \right)=n$ is equivalent in distribution to $\left(P_{\ell ,0}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[n\right]}\left(\lambda \right)={\int }_0^{\infty }\mathrm{PD}\left(\alpha \right|t)f_{\alpha }^{\left[n\right]}\left(t\right|\lambda )dt$.

(ii) 

$\left(P_{\ell ,r}\right)|N_{T_{\alpha }}\left(\lambda \right)=n,{\prod }_{i=1}^r{B}_i={\mathbf{b}}_r$ has distribution ${\mathbb{P}}_{\alpha }^{[n-r]}\left(\lambda {\mathbf{b}}_r^{-\frac{1}{\alpha }}\right),$ for $r=1,2,\dots .$

(iii) 

$\left(P_{\ell ,r}\right)|N_{T_{\alpha }}\left(\lambda \right)=n,{\prod }_{i=1}^r{B}_i={\mathbf{b}}_r$ has the same distribution as $\left(P_{\ell ,r}\right)|N_{T_{\alpha ,r}}\left(\lambda {\mathbf{b}}_r^{-\frac{1}{\alpha }}\right)=n.$

Proof. 

Statement (i) has already been established. For (ii) and equivalently (iii), we use $T_{\alpha }=T_{\alpha ,r}\times {\prod }_{i=1}^r{B}_i^{-\frac{1}{\alpha }},$ to obtain $N_{T_{\alpha }}\left(\lambda \right)=N_{T_{\alpha ,r}}\left(\lambda {\prod }_{i=1}^r{B}_i^{-\frac{1}{\alpha }}\right).$ Use (7) and (8) with $T_{\alpha ,r},$ with density $f_{\alpha ,r}\left(t\right),$ in place of $T_{\alpha },$ to conclude that $T_{\alpha ,r}|N_{T_{\alpha ,r}}\left(\lambda {\mathbf{b}}_r^{-\frac{1}{\alpha }}\right),{\prod }_{i=1}^r{B}_i={\mathbf{b}}_r$ has density $f_{\alpha }^{[n-r]}\left(t\right|\lambda {\mathbf{b}}_r^{-\frac{1}{\alpha }}).$ Then, apply $\left(P_{\ell ,r}\right)|T_{\alpha ,r}=t,N_{T_{\alpha }}\left(\lambda \right)=n,{\prod }_{i=1}^r{B}_i={\mathbf{b}}_r$ is $\mathrm{PD}\left(\alpha \right|t)$ for $\left(P_{\ell ,r}\right)\sim \mathrm{PD}(\alpha ,r).$ □

3. Results for $\mathit{n}=\mathbf{0},\mathbf{1}$

We will now focus on results for $(B_1,\dots ,B_r,T_{\alpha ,r})$, given $N_{T_{\alpha }}\left(\lambda \right)=n,$ in the cases where $n=0,1,$ and $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0).$ This is equivalent to providing more explicit distributional results than Proposition 1 for the generalized gamma and its size-biased case, where $\left(P_{\ell ,0}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[n\right]}\left(\lambda \right),$ for $n=0,1,$ subjected to the fragmentation operations in (6). We first highlight a class of random variables that will play an important role in our descriptions.

Throughout, we define ${\gamma }_{\theta }\sim \mathrm{Gamma}(\theta ,1)$ for $\theta \ge 0,$ with ${\gamma }_0:=0.$ Let $\left({\mathbf{e}}^{(\ell )}\right)$ and $\left({\gamma }_{\frac{1-\alpha }{\alpha }}^{(\ell )}\right)$ denote, respectively, iid collections of $\mathrm{exponential}\left(1\right)$ and $\mathrm{Gamma}(\frac{1-\alpha }{\alpha },1)$ random variables that are mutually independent. Use this to form iid sums ${\gamma }_{\frac{1}{\alpha }}^{\left(k\right)}:={\mathbf{e}}^{\left(k\right)}+{\gamma }_{\frac{1-\alpha }{\alpha }}^{\left(k\right)}\sim \mathrm{Gamma}(\frac{1}{\alpha },1),$ and construct increasing sums ${\Gamma }_{\alpha ,k}:={\sum }_{j=1}^k{\gamma }_{\frac{1}{\alpha }}^{\left(j\right)}\sim \mathrm{Gamma}(\frac{k}{\alpha },1)$ for $k=1,2,\dots .$

Lemma 1.

For $k=1,2,\dots ,$ set $Y_k\left(\lambda \right)=({\Gamma }_{\alpha ,k-1}+{\lambda }^{\alpha })/({\Gamma }_{\alpha ,k}+{\lambda }^{\alpha }),$ with ${\Gamma }_{\alpha ,0}=0,$ and hence $Y_1\left(\lambda \right)={\lambda }^{\alpha }/({\Gamma }_{\alpha ,1}+{\lambda }^{\alpha }).$ Then, for any $r=1,2,\dots ,$ and $\lambda >0,$ the joint density of $(Y_1\left(\lambda \right),\dots ,Y_r\left(\lambda \right))$ can be expressed as

$${\vartheta }_{\alpha ,r}^{\left[0\right]}(y_1,\dots ,y_r|\lambda )=\frac{{\lambda }^r}{{[\Gamma \left(\frac{1}{\alpha }\right)]}^r}{e}^{-{\lambda }^{\alpha }/\left({\prod }_{j=1}^r{y}_j\right)}{e}^{{\lambda }^{\alpha }}\prod _{l=1}^r{y}_l^{-\frac{(r-l+1)}{\alpha }-1}{(1-y_l)}^{\frac{1}{\alpha }-1}.$$

(9)

Furthermore, ${\lambda }^{\alpha }/{\prod }_{j=1}^r{Y}_j\left(\lambda \right)={\Gamma }_{\alpha ,r}+{\lambda }^{\alpha }.$

3.1. Results for $\left(P_{\ell ,0}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[0\right]}\left(\lambda \right),$ the Generalized Gamma Case

Let $\left({\beta }_{(\frac{1-\alpha }{\alpha },1)}^{\left(k\right)}\right)$ denote a collection of iid $\mathrm{Beta}(\frac{1-\alpha }{\alpha },1)$ variables, and independent of this, let $\left({\tau }_{\alpha }^{\left(r\right)}\left(y\right)\right)$ denote, for each fixed $y\ge 0,$ a collection of iid variables such that ${\tau }_{\alpha }^{\left(r\right)}\left(y\right)\stackrel{d}{=}{\tau }_{\alpha }\left(y\right).$ In addition, for each r, $({\beta }_{(\frac{1-\alpha }{\alpha },1)}^{\left(1\right)},\dots ,{\beta }_{(\frac{1-\alpha }{\alpha },1)}^{\left(r\right)},{\tau }_{\alpha }^{\left(r\right)}\left(\lambda \right))$ is independent of $(Y_1\left(\lambda \right),\dots ,Y_r\left(\lambda \right)).$

Proposition 2.

Consider $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,0);$ then, for each r, the joint distribution of the random variables $(B_1,\dots ,B_r,T_{\alpha ,r})|N_{T_{\alpha }}\left(\lambda \right)=0$ is equivalent component-wise and jointly to the distribution of $(B_1^{\left[0\right]}\left(\lambda \right),\cdots ,B_r^{\left[0\right]}\left(\lambda \right),T_{\alpha ,r}^{\left[0\right]}\left(\lambda \right)),$ where:

(i) 

$B_k^{\left[0\right]}\left(\lambda \right)\stackrel{d}{=}1-{\beta }_{(\frac{1-\alpha }{\alpha },1)}^{\left(k\right)}[1-Y_k\left(\lambda \right)],$ with conditional density given $Y_k\left(\lambda \right)=y_k,$

$$\frac{1-\alpha }{\alpha }{(1-b_k)}^{\frac{1-\alpha }{\alpha }-1}{(1-y_k)}^{1-\frac{1}{\alpha }}{\mathbb{I}}_{\{y_k\le b_k\le 1\}},$$

for $k=1,2,\dots$.

(ii) 

The conditional distribution of $T_{\alpha ,r}|N_{T_{\alpha }}\left(\lambda \right)=0$ is equivalent to that of

$$T_{\alpha ,r}^{\left[0\right]}\left(\lambda \right)\stackrel{d}{=}\frac{{\tau }_{\alpha }^{\left(r\right)}({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}{{({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}^{1/\alpha }}$$

where recall ${\lambda }^{\alpha }/{\prod }_{j=1}^r{Y}_j\left(\lambda \right)={\Gamma }_{\alpha ,r}+{\lambda }^{\alpha }.$

(iii) 

The conditional density of $T_{\alpha ,r}^{\left[0\right]}\left(\lambda \right)|{\prod }_{i=1}^r{Y}_i\left(\lambda \right)={\mathbf{y}}_r,$ is $f_{\alpha }^{\left[0\right]}\left(t\right|\lambda {{\mathbf{y}}_r}^{-\frac{1}{\alpha }}).$

(iv) 

Hence, $\left(P_{\ell ,r}\right)|N_{T_{\alpha }}\left(\lambda \right)=0\sim \mathbb{E}\left[{\mathbb{P}}_{\alpha }^{\left[0\right]}\left({({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}^{1/\alpha }\right)\right].$

(v) 

$(B_1^{\left[0\right]}\left(\lambda \right),\cdots ,B_r^{\left[0\right]}\left(\lambda \right),T_{\alpha ,r}^{\left[0\right]}\left(\lambda \right))|Y_1\left(\lambda \right),\dots ,Y_r\left(\lambda \right)$ are independent.

Corollary 1.

Suppose that $\left(P_{\ell ,0}\left(\lambda \right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[0\right]}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[0\right]}\left(\lambda \right)={\int }_0^{\infty }\mathrm{PD}\left(\alpha \right|t)e^{-\lambda t}{e}^{{\lambda }^{\alpha }}{f}_{\alpha }\left(t\right)dt,$ then for $r=1,2,\dots ,$

$$\left(P_{\ell ,r}\left(\lambda \right)\right)={\widehat{\mathrm{Frag}}}_{\alpha ,1-\alpha }^{\left(r\right)}\left(\left(P_{\ell ,r-1}\left(\lambda \right)\right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[0\right]}\left({({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}^{1/\alpha }\right)\right)$$

(10)

where ${\Gamma }_{\alpha ,r}={\sum }_{j=1}^r{\gamma }_{\frac{1}{\alpha }}^{\left(j\right)}\sim \mathrm{Gamma}\left(\frac{r}{\alpha }\right)$

Proof. 

This follows from statement (iv) of Proposition 2. □

The corollary shows that the fragmentation operations in (6) lead to a nested family of (mixed) normalized generalized gamma distributed mass partitions, with ${\lambda }^{\alpha }$ replaced by the random quantities ${\lambda }^{\alpha }/{\prod }_{j=1}^r{Y}_j\left(\lambda \right)={\Gamma }_{\alpha ,r}+{\lambda }^{\alpha }.$ In other words, $\left(P_{\ell ,r}\right)|N_{T_{\alpha ,0}}\left(\lambda \right)=0$ equates in distribution to the ranked masses of the random distribution function, for $v\in [0,1]$:

$$F_{\alpha }\left(v\right|{({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}^{1/\alpha })\stackrel{d}{=}\frac{{\tau }_{\alpha }\left([{\Gamma }_{\alpha ,r}+{\lambda }^{\alpha }]v\right)}{{\tau }_{\alpha }({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}.$$

Now, in order to recover ${\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta )$ for $\theta \ge 0,$ when $\left(P_{\ell ,0}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[0\right]}\left(\lambda \right),$ set, for $\theta \ge 0,$ ${\tilde{G}}_{\alpha ,\theta }\stackrel{d}{=}{G}_{\frac{\theta }{\alpha }}^{\frac{1}{\alpha }}\stackrel{d}{=}\frac{{\gamma }_{\theta }}{T_{\alpha ,\theta }},$ where $G_{\frac{\theta }{\alpha }}\sim \mathrm{Gamma}(\frac{\theta }{\alpha },1).$ When $\left(P_{\ell ,0}\left(\lambda \right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[0\right]}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[0\right]}\left(\lambda \right),$ as in Corollary 1, it follows from ([4], Proposition 21) that $\left(P_{\ell ,0}\left({\tilde{G}}_{\alpha ,\theta }\right)\right)\sim \mathrm{PD}(\alpha ,\theta ).$ Hence $(\left(P_{\ell ,r}\left({\tilde{G}}_{\alpha ,\theta }\right)\right),Z_r\left({\tilde{G}}_{\alpha ,\theta }\right);r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta ).$ It follows from Proposition 2 that, $B_k^{\left[0\right]}\left({\tilde{G}}_{\alpha ,\theta }\right)\stackrel{ind}{\sim }\mathrm{Beta}(\frac{\theta +\alpha +k-1}{\alpha },\frac{1-\alpha }{\alpha })$ for $k=1,2,\dots .$ Notably, $(Y_1\left({\tilde{G}}_{\alpha ,\theta }\right),\dots ,Y_r\left({\tilde{G}}_{\alpha ,\theta }\right))$ are independent variables, such that $1-Y_r\left({\tilde{G}}_{\alpha ,\theta }\right)\sim \mathrm{Beta}(\frac{1}{\alpha },\frac{\theta +r-1}{\alpha })$ for $r=1,2,\dots .$ When $\theta =0,$ or equivalently $\lambda =0,$ $Y_1\left(0\right)=0,$ and $1-Y_r\left(0\right)\sim \mathrm{Beta}(\frac{1}{\alpha },\frac{r-1}{\alpha })$ for $r=2,\dots .$

3.2. Results for $\left(P_{\ell ,0}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[1\right]}\left(\lambda \right)$

Proposition 3.

Consider $(\left(P_{\ell ,r}\right),Z_r;r\ge 0)|N_{T_{\alpha }}\left(\lambda \right)=1\sim {\mathrm{MLMC}}_{\mathrm{frag}}^{\left[1\right]}\left(\alpha \right|\lambda );$ then, for each r, the joint distribution of the random variables $(B_1,\dots ,B_r,T_{\alpha ,r})|N_{T_{\alpha }}\left(\lambda \right)=1$ is equivalent component-wise and jointly to the distribution of $(B_1^{\left[1\right]}\left(\lambda \right),\cdots ,B_r^{\left[1\right]}\left(\lambda \right),T_{\alpha ,r}^{\left[1\right]}\left(\lambda \right)),$ where:

(i) 

$B_1^{\left[1\right]}\left(\lambda \right)\stackrel{d}{=}{\lambda }^{\alpha }/({\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })$, where ${\gamma }_{\frac{1-\alpha }{\alpha }}\sim \mathrm{Gamma}(\frac{1-\alpha }{\alpha },1)$.

(ii) 

$B_k^{\left[1\right]}\left(\lambda \right)\stackrel{d}{=}{B}_{k-1}^{\left[0\right]}\left({({\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}^{1/\alpha }\right)$ for $k=2,3,\dots ,$ component-wise and jointly.

(iii) 

$T_{\alpha ,r}^{\left[1\right]}\left(\lambda \right)$ is equivalent in distribution to $T_{\alpha ,r}|N_{T_{\alpha }}\left(\lambda \right)=1$ and equivalent in distribution to

$$T_{\alpha ,r-1}^{\left[0\right]}\left({({\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}^{1/\alpha }\right)\stackrel{d}{=}\frac{{\tau }_{\alpha }^{(r-1)}({\Gamma }_{\alpha ,r-1}+{\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}{{({\Gamma }_{\alpha ,r-1}+{\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}^{1/\alpha }},$$

$r=1,2,\dots$.

Corollary 2.

The distributions of the components of $(\left(P_{\ell ,r}\left(\lambda \right)\right),Z_r\left(\lambda \right);r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}^{\left[1\right]}\left(\alpha \right|\lambda )$, where $\left(P_{\ell ,0}\left(\lambda \right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[1\right]}\left(\lambda \right)\right)\sim {\mathbb{P}}_{\alpha }^{\left[1\right]}\left(\lambda \right),$ for $\lambda >0,$ satisfies for $r=1,2,\dots ,$

$$\left(P_{\ell ,r}\left(\lambda \right)\right)={\widehat{\mathrm{Frag}}}_{\alpha ,1-\alpha }^{\left(r\right)}\left(\left(P_{\ell ,r-1}\left(\lambda \right)\right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[1\right]}\left({({\Gamma }_{\alpha ,r}+{\lambda }^{\alpha })}^{1/\alpha }\right)\right),$$

(11)

where $\left(P_{\ell }^{\left[1\right]}\left({({\mathbf{e}}_1+{\Gamma }_{\alpha ,r-1}+{\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}^{1/\alpha }\right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[0\right]}\left({({\Gamma }_{\alpha ,r-1}+{\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}^{1/\alpha }\right)\right)$ for ${\mathbf{e}}_1\sim \mathrm{exponential}\left(1\right)$ independent of the other variables. In this case, ${\Gamma }_{\alpha ,r}\stackrel{d}{=}{\mathbf{e}}_1+{\Gamma }_{\alpha ,r-1}+{\gamma }_{\frac{1-\alpha }{\alpha }}.$

Proof. 

$\left(P_{\ell ,r}\right)|N_{T_{\alpha }}\left(\lambda \right)=1,$ has the same distribution as $\left(P_{\ell ,r}\left(\lambda \right)\right)$ in (11), and (iii) of Proposition 3 shows that they are equivalent in distribution to $\left(P_{\ell }^{\left[0\right]}\left({({\Gamma }_{\alpha ,r-1}+{\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha })}^{1/\alpha }\right)\right).$ From ([19], Corollary 2.1, Proposition 3.2), there is the equivalence $\left(P_{\ell }^{\left[1\right]}\left({({\mathbf{e}}_1+{\lambda }^{\alpha })}^{1/\alpha }\right)\right)\stackrel{d}{=}\left(P_{\ell }^{\left[0\right]}\left(\lambda \right)\right)$ for any $\lambda \ge 0,$ yields (11). □

Now, in order to recover ${\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta )$ for $\theta >-\alpha ,$ when $\left(P_{\ell ,0}\left(\lambda \right)\right))\sim {\mathbb{P}}_{\alpha }^{\left[1\right]}\left(\lambda \right),$ use ${\widehat{G}}_{\alpha ,\theta }\stackrel{d}{=}{G}_{\frac{\theta +\alpha }{\alpha }}^{\frac{1}{\alpha }}\stackrel{d}{=}\frac{{\gamma }_{1+\theta }}{T_{\alpha ,\theta }},$ where $G_{\frac{\theta +\alpha }{\alpha }}\sim \mathrm{Gamma}(\frac{\theta +\alpha }{\alpha },1),$ and, $(\left(P_{\ell ,r}\left(\lambda \right)\right),Z_r\left(\lambda \right);r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}^{\left[1\right]}\left(\alpha \right|\lambda ).$ It follows from ([19], Corollary 2.1) that $(\left(P_{\ell ,r}\left({\widehat{G}}_{\alpha ,\theta }\right)\right),Z_r\left({\widehat{G}}_{\alpha ,\theta }\right);r\ge 0)\sim {\mathrm{MLMC}}_{\mathrm{frag}}(\alpha ,\theta ),$ for $\theta >-\alpha .$

3.3. Proofs of Propositions 2 and 3

Although the joint conditional density of $(B_1,\dots ,B_r,T_{\alpha ,r})|N_{T_{\alpha }}\left(\lambda \right)=0$ in the $\mathrm{MLMC}(\alpha ,0)$ setting can be easily obtained from ([6], p. 324), with $h\left(t\right)=e^{-\lambda t}{e}^{{\lambda }^{\alpha }},$ for clarity, we derive it here. Since $\mathbb{P}(N_{T_{\alpha }}\left(\lambda \right)=0|T_{\alpha ,r}=s,{\prod }_{i=1}^r{B}_i={\mathbf{b}}_r)=e^{-\lambda s/{{\mathbf{b}}_r}^{1/\alpha }}$, and $\mathbb{P}(N_{T_{\alpha }}\left(\lambda \right)=0)=e^{-{\lambda }^{\alpha }}$, it follows that the desired conditional density of $(B_1,\dots ,B_r,T_{\alpha ,r})|N_{T_{\alpha }}\left(\lambda \right)=0,$ can be expressed as,

$$\frac{{\alpha }^r}{{[\Gamma \left(\frac{1-\alpha }{\alpha }\right)]}^r}\prod _{i=1}^r{b}_i^{\frac{\alpha +i-1}{\alpha }-1}{(1-b_i)}^{\frac{1-\alpha }{\alpha }-1}\times s^{-r}{f}_{\alpha }\left(s\right)e^{-\lambda s/{{\mathbf{b}}_r}^{1/\alpha }}{e}^{{\lambda }^{\alpha }}.$$

(12)

Now, a joint density of $(B_1^{\left[0\right]}\left(\lambda \right),\dots ,B_r^{\left[0\right]}\left(\lambda \right),T_{\alpha ,r}^{\left[0\right]}\left(\lambda \right),Y_1\left(\lambda \right),\dots ,Y_r\left(\lambda \right))$ follows from the descriptions in Proposition 2 and Lemma 3.1 and can be expressed, for $0\le y_k\le b_k\le 1$,$k=1,\dots ,r$, as

$$e^{{\lambda }^{\alpha }}{f}_{\alpha }\left(s\right)\frac{{\lambda }^r}{{[\Gamma \left(\frac{1-\alpha }{\alpha }\right)]}^r}\prod _{k=1}^r{(1-b_k)}^{\frac{1-\alpha }{\alpha }-1}\times e^{-\lambda s/{{\mathbf{y}}_r}^{1/\alpha }}\prod _{l=1}^r{y}_l^{-\frac{(r-l+1)}{\alpha }-1},$$

(13)

for ${\mathbf{y}}_r={\prod }_{i=1}^r{y}_i.$ Proposition 2 is verified by showing that integrating over $(y_1,\dots ,y_r)$ in (13) leads to (12). This is equivalent to showing that

$${\int }_0^{b_1}\cdots {\int }_0^{b_r}{e}^{-\lambda s/{{\mathbf{y}}_r}^{1/\alpha }}\prod _{l=1}^r{y}_l^{-\frac{(r-l+1)}{\alpha }-1}d{y}_r\cdots d{y}_1={\alpha }^r{\lambda }^{-r}{s}^{-r}{e}^{-\lambda s/{{\mathbf{b}}_r}^{1/\alpha }}\prod _{i=1}^r{b}_i^{\frac{i-1}{\alpha }}.$$

which follows by elementary calculations involving the change of variable $v_i=y_i^{-1/\alpha },$ for $i=1,\dots ,r$ and exponential integrals. Now, to establish Proposition 3, first note that since $\mathbb{P}(N_{T_{\alpha }}\left(\lambda \right)=1|T_{\alpha ,1}=s,B_1=b_1)=\lambda s{b}_1^{-\frac{1}{\alpha }}{e}^{-\lambda s/b_1^{\frac{1}{\alpha }}}$, and $\mathbb{P}(N_{T_{\alpha }}\left(\lambda \right)=1)=\alpha {\lambda }^{\alpha }{e}^{-{\lambda }^{\alpha }},$ the joint density of $B_1,T_{\alpha ,1}|N_{T_{\alpha }}\left(\lambda \right)=1$ can be expressed as

$$\frac{{\lambda }^{1-\alpha }}{\Gamma \left(\frac{1-\alpha }{\alpha }\right)}{b}_1^{-\frac{1}{\alpha }}{(1-b_1)}^{\frac{1-\alpha }{\alpha }-1}\times e^{-\lambda s/{b_1}^{1/\alpha }}{e}^{{\lambda }^{\alpha }}{f}_{\alpha }\left(s\right).$$

(14)

Hence, the conditional density of $B_1|N_{T_{\alpha }}\left(\lambda \right)=1$ can be expressed as,

$$\frac{{\lambda }^{1-\alpha }}{\Gamma \left(\frac{1-\alpha }{\alpha }\right)}{b}_1^{-\frac{1}{\alpha }}{(1-b_1)}^{\frac{1-\alpha }{\alpha }-1}\times e^{-{\lambda }^{\alpha }/b_1}{e}^{{\lambda }^{\alpha }}.$$

(15)

which corresponds to $B_1^{\left[1\right]}\left(\lambda \right)\stackrel{d}{=}{\lambda }^{\alpha }/({\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha }),$ verifying statement (i) of Proposition 3. Refs. (14) and (15) show that $T_{\alpha ,1}|N_{T_{\alpha }}\left(\lambda \right)=1,B_1=b_1$ is $f_{\alpha }^{\left[0\right]}\left(s\right|\lambda b_1^{-\frac{1}{\alpha }})$, which leads to $\left(P_{\ell ,1}\right)|N_{T_{\alpha }}\left(\lambda \right)=1,B_1=b_1$ having distribution ${\mathbb{P}}_{\alpha }^{\left[0\right]}\left(\lambda b_1^{-\frac{1}{\alpha }}\right).$ This agrees with statement (ii) of Proposition 1, with $n=r=1.$ Using ${\lambda }^{\alpha }/B_1\left(\lambda \right)\stackrel{d}{=}{\gamma }_{\frac{1-\alpha }{\alpha }}+{\lambda }^{\alpha }$ and applying Proposition 2 starting with $\left(P_{\ell ,1}\right)|N_{T_{\alpha }}\left(\lambda \right)=1,B_1=b_1$ subject to (6) concludes the proof of Proposition 3.