2. $\alpha$ -stable Poisson–Kingman distributions and Gibbs partitions

Letting $(\Delta_{k})_{{k\ge 1}}$ , with $\Delta_{1}>\Delta_{2}>\cdots$ , denote the ranked jumps of an $\alpha$ -stable subordinator, say $\mathbf{T}_{\alpha}\,:\!=\,({T}_{\alpha}(t)\colon t\ge 0)$ of index $\alpha\in(0,1)$ , with $T_{\alpha}(0)=0$ and $T_{\alpha}(1)\,:\!=\,T_{\alpha}=\sum_{k=1}^{\infty}\Delta_{k}$ a positive stable random variable having Laplace transform $\mathbb{E}[{\text e}^{-\lambda T_{\alpha}}]={\text e}^{-\lambda^{\alpha}}$ and density denoted as $f_{\alpha}(t)$ . In addition, let $(L_{t}\colon t \ge 0)$ denote the inverse of $\mathbf{T}_{\alpha}$ , which corresponds to the local-time process of a strong Markov process starting at 0, for instance a Bessel process of dimension $ 2 - 2\alpha $ starting at zero. As in [Reference Pitman and Yor36], we may use the notation L(t) in place of $ L_{t} $ for notational convenience. The local time up to time 1 satisfies $ L(1) \,:\!=\, L_{1} \overset{\mathrm{d}}{=} T^{-\alpha} \sim \mathrm{ML}(\alpha,0) $ , having a Mittag-Leffler distribution with density denoted as $ g_{\alpha} $ . Hereafter, we will simply say $ L(1) \,:\!=\, L_{1} $ is the local time at 1. That is, $T_{\alpha}(s)=\inf\{t\colon L_{t}>s\}$ , $s\ge 0$ , is the corresponding inverse local-time process. We have the following important scaling identity (see [Reference Pitman and Yor36]):

(2.1) \begin{equation} L_{1}\overset{\mathrm{d}} = \frac{L_{t}}{t^{\alpha}}\overset{\mathrm{d}} = \frac{s}{[{{T_{\alpha}(s)]}^{\alpha}}}\overset{\mathrm{d}} = T^{-\alpha}_{\alpha}.\end{equation}

Then $(P_{k}\,:\!=\,\Delta_{k}/T_{\alpha})_{k\ge 1}\sim \mathrm{PD}(\alpha,0)$ [Reference Pitman and Yor37], a two-parameter $(\alpha,0)$ Poisson–Dirichlet distribution, where as an example $\mathrm{PD}\big(\frac12,0\big)$ corresponds to the distribution of the lengths of excursions of a Brownian motion on [0,1]. Furthermore, $F(u)\,:\!=\,F_{\alpha}(u)\overset{\mathrm{d}}=T_{\alpha}(u)/T_{\alpha}(1)$ for $u\in[0,1]$ . Another important property, relevant to species sampling or species abundance models, is that $n^{-\alpha}K_{n}$ converges almost surely to $T^{-\alpha}_{\alpha}$ , where in this context $T^{-\alpha}_{\alpha}$ is referred to as the $\alpha$ -diversity of $\Pi_{\infty}$ , as in [Reference Pitman32, Reference Pitman33].

To construct the larger class of distributions, let $\mathrm{PD}(\alpha\mid t)$ denote the conditional distribution of $ (P_{k})_{k \geq 1} $ given $T_{\alpha}=t$ (or $L_{1}=t^{-\alpha})$ and, for a non-negative function h(t) satisfying $\mathbb{E}[h(T_{\alpha})]=1$ , mix $\mathrm{PD}(\alpha\mid t)$ over the probability measure $\gamma(\mathrm{d}t)/{\mathrm{d}} t=h(t)f_{\alpha}(t)$ to obtain $(P_k)_{k\ge 1} \sim \mathrm{PK}_{\alpha}(h\cdot f_{\alpha})$ , where

\begin{equation*} \mathrm{PK}_{\alpha}(h\cdot f_{\alpha}) = \int_{0}^{\infty}\mathrm{PD}(\alpha\mid t)h(t)f_{\alpha}(t)\,{\mathrm{d}} t = \int_{0}^{\infty}\mathrm{PD}\big(\alpha\mid s^{-{1}/{\alpha}}\big)h\big(s^{-{1}/{\alpha}}\big)g_{\alpha}(s)\,{\mathrm{d}} s.\end{equation*}

It follows from [Reference Ho, James and Lau20, Reference Ho, James and Lau21] that the conditional distribution of $T_{\alpha}\mid K_{n}=k$ equates to the distribution of the variable $Y^{(n-k\alpha)}_{\alpha,k\alpha}$ , with density $f^{(n-k\alpha)}_{\alpha,k\alpha}(t)$ , such that (pointwise), as in [Reference Ho, James and Lau21, (2.13), p. 323],

(2.2) \begin{equation} Y^{(n-k\alpha)}_{\alpha,k \alpha}\overset{\mathrm{d}} = \frac{T_{\alpha,k\alpha}}{B_{k\alpha,n-k\alpha}}=\frac{T_{\alpha,n}}{B^{1/\alpha}_{k,{n}/{\alpha}-k}},\end{equation}

where the variables in each ratio are independent, and throughout $B_{a,b}$ denotes a $\mathrm{Beta}(a,b)$ random variable. Hence, the EPPF of the general case is given in (1.1) with ${\Psi}^{[\alpha]}_{n,k}$ expressed as ${\Psi}^{[\alpha]}_{n,k} =\mathbb{E}[h(T_{\alpha})\mid K_{n}=k]=\mathbb{E}\big[h\big(Y^{(n-k\alpha)}_{\alpha,k\alpha}\big)\big]$ . It is noteworthy that (2.2) indicates that Mittag-Leffler variables play a role in the general $\alpha$ class of Gibbs partitions. We can also use this to verify the recursion (see [Reference Gnedin and Pitman15] and [Reference Ho, James and Lau21, Lemma 2.1])

\begin{align*}{\Psi}^{[\alpha]}_{n,k}=\frac{k\alpha}{n}{\Psi}^{[\alpha]}_{n+1,k+1}+\bigg(1-\frac{k\alpha}{n}\bigg){\Psi}^{[\alpha]}_{n+1,k}.\end{align*}

Remarkably, for the number of blocks $K_{n}$ having the probability mass function ${\Psi}^{[\alpha]}_{n,k} \mathbb{P}^{(n)}_{\alpha,0}(k)$ , it follows from [Reference Pitman32, Proposition 13] that as $n \rightarrow \infty$ , $n^{-\alpha} K_{n}$ converges almost surely to $T^{-\alpha}$ , which is the $\alpha$ -diversity under $\mathrm{PK}_{\alpha}(h \cdot f_{\alpha})$ with density $h(s^{-{1}/{\alpha}}) g_{\alpha}(s)$ , regardless of how complex ${\Psi}^{[\alpha]}_{n,k}$ may appear.

3. Fragmentation operator $\mathbf{FRAG}_{\boldsymbol{\alpha},-\boldsymbol{\beta}}$

For $0 \leq \beta < \alpha$ , we define the $\mathrm{PD}(\alpha,-\beta)$ fragmentation operator on $\mathcal{P}_{\infty}$ (and also $\Pi_{\infty}$ ) as in [Reference Pitman31], and more generally in [Reference Pitman33, p. 108]. We refer to the operation on $\Pi_{\infty}$ as the $\mathrm{CRP}(\alpha,-\beta)$ fragmentation operator. In both cases, we use $\mathrm{FRAG}_{\alpha,-\beta}$ to denote this operation in the respective contexts.

3.1. $\mathbf{PD}(\boldsymbol{\alpha},-\boldsymbol{\beta})$ fragmentation on mass partitions

Let $\big(\hat{Q}^{(k)}_{j}\big)_{j \geq 1} \sim \mathrm{PD}(\alpha,-\beta))$ for $k=1,2,\ldots$ denote a countably infinite collection of independent elements in $\mathcal{P}_{\infty}$ , each having a $\mathrm{PD}(\alpha,-\beta)$ distribution. Independent of this, select $\mathbf{V} \,:\!=\, (V_{k})_{k \geq 1} \in \mathcal{P}_{\infty}$ . A $\mathrm{PD}(\alpha,-\beta)$ fragmentation operation first shatters each mass $V_{k}$ by an independent mass partition $\big(\hat{Q}^{(k)}_{j}\big)_{j \geq 1} \sim \mathrm{PD}(\alpha,-\beta)$ such that $V_{k}\big(\hat{Q}^{(k)}_{j}\big)_{j \geq 1} = \big(V_{k}\hat{Q}^{(k)}_{1}, V_{k}\hat{Q}^{(k)}_{2}, \ldots\big)$ for $k=1,2,\ldots$ The collection is then arranged in ranked order as an element in $\mathcal{P}_{\infty}$ . The $\mathrm{FRAG}_{\alpha,-\beta}$ operator is defined as follows, with Rank denoting the operation corresponding to the ranked rearrangement of masses:

(3.1) \begin{equation} \mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V}) = \mathrm{Rank}\big(V_{k} \hat{Q}^{(k)}_{j}, j \geq 1, k \geq 1\big) \in \mathcal{P}_{\infty}.\end{equation}

3.2. $\mathbf{CRP}(\alpha,-\beta)$ fragmentation on integer partitions

The corresponding $(\alpha,-\beta)$ fragmentation operation on the space of integer partitions $\Pi_{\infty}$ for a given partition $\Pi_{n}=\{C_{1},\ldots,C_{K_{n}}\}$ is generated from the corresponding $\mathbf{V}$ . This operation shatters each block $C_{j}$ by an independent $\mathrm{CRP}(\alpha,-\beta)$ partition of the set $C_{j}$ , with EPPF $p_{\alpha,-\beta}$ arising as a special case of $p_{\alpha,\theta}$ specified by (1.2). For given $C_{j}$ , this results in a $\mathrm{CRP}(\alpha,-\beta)$ partition of $C_{j}$ , denoted as $\big\{\tilde{C}_{1,j},\ldots, \tilde{C}_{K^{(j)}_{|C_{j}|},j}\big\}$ , where the number of unique blocks $K^{(j)}_{|C_{j}|} \leq |C_{j}|$ corresponds to that of a $\mathrm{CRP}(\alpha,-\beta)$ partition of $C_{j}$ . The $\mathrm{FRAG}_{\alpha,-\beta}$ operation on $\Pi_{n}=\{C_{1},\ldots,C_{K_{n}}\}$ results in a partition of [n] with $\sum_{j=1}^{K_{n}} K^{(j)}_{|C_{j}|} \leq n$ distinct blocks formed by the collection

(3.2) \begin{equation} \big\{\tilde{C}_{i,j}, i=1,\ldots, K^{(j)}_{|C_{j}|}, j=1,\ldots,K_{n}\big\},\end{equation}

arranged according to the least element.

In the case where $\mathbf{V} \sim \mathrm{PD}(\beta,\theta)$ , [Reference Pitman31] shows that $\tilde{\mathbf{V}}=\mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V}) \sim \mathrm{PD}(\alpha,\theta)$ . Conversely, as in [Reference Pitman31, Reference Pitman33], applying an independent $\mathrm{PD}(\beta/\alpha,\theta/\alpha)$ coagulation operation to the fragmented process $\tilde{\mathbf{V}}$ reverses this operation. The corresponding duality on integer partitions $\Pi_{\infty}$ is described in more detail in [Reference Pitman31, Theorem 12]. For a general treatment of these notions, see [Reference Bertoin3].

3.3. $\mathbf{PD}(\alpha,-\beta)$ fragmentation of $\mathbf{PK}_{\beta}(h\cdot f_{\beta})$ mass partitions

We now address the question of the resulting distribution from the same independent fragmentation operation, $\mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V})$ , when $\mathbf{V}\,:\!=\,(V_{k})_{k \geq 1} \sim \mathrm{PK}_{\beta}(h \cdot f_{\beta})$ , where for clarity h(t) is chosen with respect to $T_{\beta}$ , having density $f_{\beta}$ such that $\mathbb{E}[h(T_{\beta})]=1$ . In order to achieve our results, we work with independent stable subordinators $\mathbf{T}_{\alpha}\,:\!=\,(T_{\alpha}(t),t\ge 0)$ and $\mathbf{T}_{{\beta}/{\alpha}}\,:\!=\,(T_{{\beta}/{\alpha}}(t),t\ge 0)$ generating independent elements of $\mathcal{P}_{\infty}$ , $\mathbf{V}_{\alpha}\sim\mathrm{PD}(\alpha,0)$ and $\mathbf{Q}_{{\beta}/{\alpha}}\sim \mathrm{PD}({\beta}/{\alpha},0)$ respectively. The corresponding independent local-time processes starting at zero are $(L_{\alpha}(t),t\ge 0)$ and $(L_{{\beta}/{\alpha}}(t),t\ge 0)$ , satisfying (2.1), with local times at 1, denoted respectively as $L_{1,\alpha}\,:\!=\,L_{\alpha}(1)\overset{\mathrm{d}}=T^{-\alpha}_{\alpha}$ and $L_{1,{\beta}/{\alpha}}\,:\!=\,L_{{\beta}/{\alpha}}(1)\overset{\mathrm{d}}=T^{-{\beta}/{\alpha}}_{{\beta}/{\alpha}}$ , playing the role of $L_{1}$ , as we have described, and otherwise following the more detailed description in [Reference Bertoin and Pitman5] as it relates to the special case of the duality [Reference Pitman31, Theorem 14 and Corollary 15]. The mass partition $\mathbf{V}_{\beta}\sim \mathrm{PD}(\beta,0)$ is formed by the independent coagulation $\mathbf{V}_{\beta}=\mathrm{PD}({\beta}/{\alpha},0)-\mathrm{COAG}(\mathbf{V}_{\alpha})$ , which, as in [Reference Bertoin3–Reference Bertoin and Pitman5] (for more details see the description in [Reference Pitman33, Lemma 5.18, p. 114–115] in relation to the Bolthausen–Sznitman coalescent [Reference Bolthausen and Sznitman6]), corresponds to the ranked normalized jumps formed by the composition of the processes.

(3.3) \begin{equation} (F_{\beta}(u) = F_{\alpha}(F_{{\beta}/{\alpha}}(u)), u\in[0,1]) \overset{\mathrm{d}} = \bigg(\frac{T_{\alpha}(T_{{\beta}/{\alpha}}(u))}{T_{\alpha}(T_{{\beta}/{\alpha}}(1))},u\in[0,1]\bigg),\end{equation}

and has local time starting at 0 up to time 1 $L_{1,\beta}\,:\!=\,L_{\beta}(1)=L_{{\beta}/{\alpha}}(L_{1,\alpha})\overset{{\mathrm{d}}}=T^{-\beta}_{\beta}$ , with inverse local time up to time 1 $T_{\beta}\,:\!=\,T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1))$ , with density $f_{\beta}$ . Conversely, $\mathbf{V}_{\alpha}=\mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V}_{\beta})\sim \mathrm{PD}(\alpha,0)$ .

Remark 3.1. We will write $F_{\alpha}\circ F_{{\beta}/{\alpha}}$ to indicate the composition of functions $F_{\alpha}(F_{{\beta}/{\alpha}}(u))$ for $u\in[0,1]$ .

Remark 3.2. There is the well-known distributional equivalence $T_{\beta} \overset{{\mathrm{d}}}= T_{\alpha}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}} \overset{{\mathrm{d}}}= T_{{\beta}/{\alpha}}\times T^{{\alpha}/{\beta}}_{\alpha}$ . However, in the case of interpretation of the $\mathrm{PD}({\beta}/{\alpha},0)$ coagulation, as in (3.3), the order matters and thus we will only use $T_{\beta}\overset{{\mathrm{d}}}=T_{\alpha}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}}$ .

Applying the scaling property in (2.1), it follows that, for any $s>0$ , $[L_{{\beta}/{\alpha}}(s^{-\alpha})]^{-{1}/{\beta}}\overset{{\mathrm{d}}}=T^{{1}/{\alpha}}_{{\beta}/{\alpha}}s$ , and also, using $L_{1,\alpha}\overset{{\mathrm{d}}}=T^{-\alpha}_{\alpha}$ independent of the process $(L_{{\beta}/{\alpha}}(t),t\ge 0)$ , we have

(3.4) \begin{equation} \omega^{(y)}_{{\beta}/{\alpha},\beta}(s) \,:\!=\, \frac{\mathbb{P}\big([L_{{\beta}/{\alpha}}(L_{1,\alpha})]^{-{1}/{\beta}}\in{\mathrm{d}} y\mid L^{-{1}/{\alpha}}_{1,\alpha} = s \big)} {\mathbb{P}\big(L^{-{1}/{\beta}}_{1,\beta}\in {\mathrm{d}} y\big)} = \frac{\alpha y^{\alpha-1}f_{{\beta}/{\alpha}}({(y/s)}^{\alpha})}{s^{\alpha}f_{\beta}(y)}\end{equation}

such that the conditional distribution of $L_{1,\alpha}\mid L_{1,\beta}$ may be expressed in terms of that of the transformed variable $L^{-{1}/{\alpha}}_{1,\alpha}\mid L^{-{1}/{\beta}}_{1,\beta}$ as

(3.5) \begin{equation} \mathbb{P}\Big(L^{-{1}/{\alpha}}_{1,\alpha}\in{\mathrm{d}} s\mid[L_{{\beta}/{\alpha}}(L_{1,\alpha})]^{-{1}/{\beta}} = y\Big)/{\mathrm{d}} s = \omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f_{\alpha}(s),\end{equation}

which is equivalent to the conditional density of $T_{\alpha}$ given $T_{\alpha}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}}=y$ . We now state our first result.

Theorem 3.1. Let $\mathbf{V}\sim \mathrm{PK}_{\beta}(h\cdot f_{\beta})$ with local time at 1, say $L_{1,\mathbf{V}}$ , having density $h(x^{-{1}/{\beta}})g_{\beta}(x)$ . For any choice of $0<\beta<\alpha<1$ , let $\mathrm{FRAG}_{\alpha,-\beta}(\cdot)$ denote a $\mathrm{PD}(\alpha,-\beta)$ fragmentation operator independent of $\mathbf{V}$ , as defined in (3.1). Then:

1. (i) $\tilde{\textbf{V}}\,:\!=\,\mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V}) \sim \mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ , where

   \begin{align*} \tilde{h}_{{\beta}/{\alpha}}(s) \,:\!=\, \mathbb{E}_{{\beta}/{\alpha}}\big[h\big(sT^{{1}/{\alpha}}_{{\beta}/{\alpha}}\big)\big] = \int_{0}^{\infty}h(sy^{{1}/{\alpha}})f_{{\beta}/{\alpha}}(y)\,{\mathrm{d}} y. \end{align*}
   That is, it has a local time at time 1, say $L_{1,\tilde{\mathbf{V}}}$ , with density $\tilde{h}_{{\beta}/{\alpha}}(z^{-{1}/{\alpha}})g_{\alpha}(z)$ .

2. (ii) The conditional distribution of $\tilde{\mathbf{V}}\mid L_{1,\mathbf{V}}=y^{-\beta}$ is equivalent to the distribution of $\mathbf{V}_{\alpha}\mid L_{1,\beta}=y^{-\beta}$ , where $\mathbf{V}_{\alpha}=\mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V}_{\beta})\sim \mathrm{PD}(\alpha,0)$ , which is

   (3.6) \begin{equation} \mathrm{PD}_{\alpha|\beta}(\alpha\mid y) \,:\!=\, \int_{0}^{\infty}\mathrm{PD}(\alpha\mid s)\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f_{\alpha}(s)\,{\mathrm{d}} s = \mathrm{PK}_{\alpha}\big(\omega^{(y)}_{{\beta}/{\alpha},\beta}\cdot f_{\alpha}\big) \end{equation}
   for $\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)$ defined in (3.4).

3. (iii) For clarity, when $\mathbf{V}_{\beta}(y)$ is equivalent in distribution to $\mathbf{V}_{\beta}\mid T_{\beta}=y\sim \mathrm{PD}(\beta\mid y)$ , it follows that $\mathrm{FRAG}_{\alpha,-\beta}(\mathbf{V}_{\beta}(y))\sim\mathrm{PD}_{\alpha|\beta}(\alpha\mid y)$ as in (3.6).

Proof. Let $\mathbb{\hat{E}}^{(\beta,0)}_{(\alpha,-\beta)}$ denote the expectation with respect to the joint law of $(\mathbf{V}_{\beta},((\hat{Q}^{(k)})_{j\ge 1}$ , $ k\ge 1))$ where $\textbf{V}_{\beta}\sim \mathrm{PD}(\beta,0)$ with local time at 1 $L_{1,\beta}=L_{{\beta}/{\alpha}}(L_{1,\alpha})$ , with density $g_{\beta}(x)$ , and, independent of this, $((\hat{Q}^{(k)})_{j\ge 1}, k\ge 1)$ are iid $\mathrm{PD}(\alpha,-\beta)$ mass partitions. Consider $\textbf{V}\sim \mathrm{PK}_{\beta}(h\cdot f_{\beta})$ . The distribution of $\tilde{\textbf{V}}=\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V})$ is characterized, for a measurable function $\Phi$ , by

(3.7) \begin{equation} \mathbb{E}[\Phi(\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V}))] = \mathbb{\hat{E}}^{(\beta,0)}_{(\alpha,-\beta)} \Big[\Phi(\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V}_{\beta}))h\big(L^{-{1}/{\beta}}_{1,\beta}\big)\Big]. \end{equation}

But, from [Reference Bertoin and Pitman5, Reference Pitman31], for $\textbf{V}_{\alpha}\sim \mathrm{PD}(\alpha,0)$ this is equivalent to

(3.8) \begin{equation} \mathbb{E}\Big[\Phi(\textbf{V}_{\alpha})h({[L_{{\beta}/{\alpha}}(L_{1,\alpha})]}^{-{1}/{\beta}})\Big]. \end{equation}

Now use (3.4) and (3.5) to write the joint density of $L^{-{1}/{\alpha}}_{1,\alpha}$ and ${[L_{{\beta}/{\alpha}}(L_{1,\alpha})]}^{-{1}/{\beta}}$ for the respective arguments (s,y) as $\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f_{\alpha}(s)f_{\beta}(y)$ . Then, conditioning on the joint event $L^{-{1}/{\alpha}}_{1,\alpha}=s$ and ${[L_{{\beta}/{\alpha}}(L_{1,\alpha})]}^{-{1}/{\beta}}=y$ , we can express (3.8) as

(3.9) \begin{equation} \int_{0}^{\infty}\int_{0}^{\infty} \mathbb{E}\big[\Phi(\textbf{V}_{\alpha})\mid L^{-{1}/{\alpha}}_{1,\alpha}=s,\, {[L_{{\beta}/{\alpha}}(L_{1,\alpha})]}^{-{1}/{\beta}}=y\big] h(y)\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f_{\alpha}(s)f_{\beta}(y)\,{\mathrm{d}} s\,{\mathrm{d}} y. \end{equation}

It follows that, since $(L_{{\beta}/{\alpha}}(t), t \ge 0)$ is independent of $\mathbf{V}_{\alpha}$ , $\mathbf{V}_{\alpha} \mid L^{-{1}/{\alpha}}_{1,\alpha}=s,{[L_{{\beta}/{\alpha}}(L_{1,\alpha})]}^{-{1}/{\beta}}=y$ has distribution $\mathrm{PD}(\alpha\mid s)$ . Using this and (3.9), it follows that the expectation in (3.8) can be expressed as

(3.10) \begin{equation} \int_{0}^{\infty}\bigg[\int_{0}^{\infty} \mathbb{E}\big[\Phi(\textbf{V}_{\alpha})\mid T_{\alpha}=s\big]\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f_{\alpha}(s)\,{\mathrm{d}} s\bigg] h(y)f_{\beta}(y)\,{\mathrm{d}} y. \end{equation}

Alternatively, it can also be expressed as $\int_{0}^{\infty}\mathbb{E}[\Phi(\textbf{V}_{\alpha})\mid T_{\alpha}=s]\tilde{h}_{{\beta}/{\alpha}}(s)f_{\alpha}(s)\,{\mathrm{d}} s$ for $\tilde{h}_{{\beta}/{\alpha}}(s)\,:\!=\,\mathbb{E}_{{\beta}/{\alpha}}\big[h\big(sT^{{1}/{\alpha}}_{{\beta}/{\alpha}}\big)\big] = \int_{0}^{\infty}\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)h(y)f_{\beta}(y)\,{\mathrm{d}} y$ , yielding the results in (i) and (ii). Now comparing $\mathbb{\hat{E}}^{(\beta,0)}_{(\alpha,-\beta)} \big[\Phi(\mathrm{FRAG}_{\alpha,-\beta}(V_{\beta}))h\big(L^{-{1}/{\beta}}_{1,\beta}\big)\big]$ in (3.7) with the equivalent expression in (3.10), it follows that, for almost all y,

\begin{align*} \mathbb{\hat{E}}^{(\beta,0)}_{(\alpha,-\beta)}\big[\Phi(\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V}_{\beta}))\mid T_\beta=y\big] = \int_{0}^{\infty}\mathbb{E}[\Phi(\textbf{V}_{\alpha})\mid T_{\alpha}=s] \omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f_{\alpha}(s)\,{\mathrm{d}} s, \end{align*}

and, noting the independence of the $\mathrm{FRAG}_{\alpha,-\beta}$ operator relative to $\textbf{V}_{\beta}$ , this leads to the result in (iii).

4. Duality via dependent coagulation

We now describe how to construct dependent coagulations to complete the dual process of recovering $\mathbf{V}\sim \mathrm{PK}_{\beta}(h\cdot f_{\beta})$ from the coagulation of $\tilde{\mathbf{V}}=\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V})\sim \mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ by some mass partition $\mathbf{Q}\,:\!=\,(Q_{\ell})_{\ell\ge 1}\in \mathcal{P}_{\infty}$ . Our results show how specification of h(t) leads to a prescription for identifying the laws of $\mathbf{V}$ , $\tilde{\mathbf{V}}$ , and $\mathbf{Q}$ without guesswork.

Recall that for the independent mass partitions $(\mathbf{V}_{\alpha},\mathbf{Q}_{{\beta}/{\alpha}})$ described in (3.3), the process of coagulation yields an inverse local time at 1 for $\mathbf{V}_{\beta}$ to be $T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1))\overset{{\mathrm{d}}}=T_{\alpha}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}}$ . For $\tilde{\mathbf{V}}$ as described above, we consider the dependent pair $(\tilde{\mathbf{V}},\mathbf{Q})$ with joint law, say $\mathrm{P}^{{\beta}/{\alpha}}_{\alpha}(h)$ , characterized by, for some generic measurable function $\Phi$ ,

(4.1) \begin{equation} \mathbb{E}[\Phi(\tilde{\mathbf{V}},\mathbf{Q})] = \mathbb{E}^{({\beta}/{\alpha},0)}_{(\alpha,0)} \big[\Phi(\mathbf{V}_{\alpha},\mathbf{Q}_{{\beta}/{\alpha}})h(T_{\alpha}(T_{{\beta}/{\alpha}}(1)))\big],\end{equation}

with $\mathbb{E}[h(T_{\beta}(1))]=\mathbb{E}[h(T_{\alpha}(T_{{\beta}/{\alpha}}(1)))]=1$ , and the notation $\mathbb{E}^{({\beta}/{\alpha},0)}_{(\alpha,0)}$ referring to an expectation evaluated under the joint law of the independent $\mathrm{PD}(\alpha,0)$ and $\mathrm{PD}(\beta/\alpha,0)$ distributions. We use this for clarity, but will suppress it when it is clear we are referring to such variables. Equivalently, by conditioning and scaling properties, the joint law of $(\tilde{\mathbf{V}},\mathbf{Q})$ is given by

(4.2) \begin{equation} \mathrm{P}^{{\beta}/{\alpha}}_{\alpha}(h) \,:\!=\, \int_{0}^{\infty}\int_{0}^{\infty} \mathrm{PD}(\alpha\mid s)\mathrm{PD}({\beta}/{\alpha}\mid y)h(sy^{{1}/{\alpha}}) f_{{\beta}/{\alpha}}(y)f_{\alpha}(s)\,{\mathrm{d}} y\,{\mathrm{d}} s.\end{equation}

In addition, for collections of iid $\mathrm{Uniform}[0,1]$ variables $((\tilde{U}_{k}),(U_{\ell}))$ independent of $(\tilde{\textbf{V}},\textbf{Q})$ , define random distribution functions (exchangeable bridges), for $y\in[0,1]$ ,

(4.3) \begin{equation} F_{\tilde{\mathbf{V}}}(y) = \sum_{k=1}^{\infty}\tilde{V}_{k}1_{\{\tilde{U}_{k}\leq y\}}, \qquad F_{\mathbf{Q}}(y) = \sum_{\ell=1}^{\infty}Q_{\ell}1_{\{U_{\ell}\leq y\}}.\end{equation}

Proof. We first recall from (3.3) that under independent $\mathrm{PD}(\alpha,0)$ and $\mathrm{PD}({\beta}/{\alpha},0)$ laws, the bridge $F_{\beta}\,:\!=\,F_{\alpha}\circ F_{{\beta}/{\alpha}}$ follows the law of a $\mathrm{PD}(\beta,0)$ -bridge with inverse local time at 1, $T_{\beta}\,:\!=\,T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1))$ . Hence, under the joint law of $(\tilde{\textbf{V}},\textbf{Q})$ specified by (4.1), it follows that, for $F_{\mathbf{V}}\,:\!=\,F_{\tilde{\mathbf{V}}}\circ F_{\mathbf{Q}}$ ,

\begin{align*} \mathbb{E}[\Phi(F_{\tilde{\mathbf{V}}}\circ F_{\mathbf{Q}})] = \mathbb{E}\big[\Phi(F_{\alpha}\circ F_{{\beta}/{\alpha}})h(T_{\alpha}(T_{{\beta}/{\alpha}}(1)))\big] = \mathbb{E}[\Phi(F_{\beta})h(T_{\beta}(1))], \end{align*}

showing that $F_{\textbf{V}}$ is a $\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ -bridge and thus $\textbf{V}\sim \mathrm{PK}_{\beta}(h\cdot f_{\beta})$ in statement (i). Statements (ii) and (iii) follow from straightforward usage of (4.2).

5. Gibbs partitions of [n] derived from $\mathbf{FRAG}_{\alpha,-\beta}$

Recall from [Reference Pitman32, Reference Pitman33] that when $\textbf{V}_{\beta}\sim \mathrm{PD}(\beta,0)$ , $\textbf{V}_{\beta}\mid L_{1,\beta} = y^{-\beta}$ is equivalent in distribution to $\textbf{V}_{\beta}\mid T_{\beta}=y\sim\mathrm{PD}(\beta\mid y)$ and has the associated Gibbs partition of [n] described by the $\mathrm{PD}(\beta\mid y)$ -EPPF

(5.1) \begin{equation} p_{\beta}(n_{1},\ldots,n_{k}\mid y) \,:\!=\, \frac{f^{(n-k\beta)}_{\beta,k\beta}(y)}{f_{\beta}(y)} p_{\beta}(n_{1},\ldots,n_{k}),\end{equation}

where, as in [Reference Ho, James and Lau20, Reference Ho, James and Lau21],

\begin{align*}\frac{f^{(n-k\beta)}_{\beta,k\beta}(y)}{f_{\beta}(y)} = \mathbb{G}^{(n,k)}_{\beta}(y)\frac{{\beta}^{1-k}\Gamma(n)}{\Gamma(k)},\end{align*}

with, from [Reference Gnedin and Pitman15, Reference Pitman32, Reference Pitman33],

(5.2) \begin{equation}\mathbb{G}_{\beta}^{(n,k)}(t) =\frac{\beta^{k}t^{-n}}{\Gamma(n-k\beta)f_{\beta}(t)}\bigg[\int_{0}^{t}f_{\beta}(v)(t-v)^{n-k\beta-1}\,{\mathrm{d}} v\bigg],\end{equation}

and $f^{(n-k\beta)}_{\beta,k\beta}(y)$ being the conditional density of $T_{\beta}\mid K^{[\beta]}_{n}=k$ corresponding to a random variable denoted as $Y^{n-k\beta}_{\beta,k\beta}$ , as otherwise described in (2.2) with $\beta$ in place of $\alpha$ . Note, furthermore, as in [Reference Ho, James and Lau21], this means that $T_{\beta}\,:\!=\,T_{\alpha}(T_{{\beta}/{\alpha}}(1))\overset{\textrm{d}}=Y^{(n-K^{[\beta]}_{n}\beta)}_{\beta,K^{[\beta]}_{n}\beta}$ for $K^{[\beta]}_{n}\sim\mathbb{P}^{(n)}_{\beta,0}(k)$ . We use these facts to obtain interesting expressions for $\alpha$ -Gibbs partitions equivalent to those arising from the $\mathrm{FRAG}_{\alpha,-\beta}$ operator. In particular, in Section 5.1 we provide a remarkable description of the corresponding EPPF for mass partitions equivalent in distribution to

(5.3) \begin{equation} \mathrm{FRAG}_{\alpha,-\beta}(\textbf{V}_{\beta}(y)) \sim \mathrm{PD}_{\alpha|\beta}(\alpha\mid y) = \mathrm{PK}_{\alpha}\big(\omega^{(y)}_{{\beta}/{\alpha},\beta}\cdot f_{\alpha}\big)\end{equation}

for $\textbf{V}_{\beta}(y)\sim \mathrm{PD}(\beta\mid y)$ , as described in the result in Theorem 3.1(iii).

5.1. Gibbs partitions of [n] of $V_{\alpha}\mid L_{1,\beta}$ , equivalently of $\mathrm{FRAG}_{\alpha,-\beta}(V)\mid L_{1,V}$

Recall from Theorem 3.1 that the distribution of $\textbf{V}_{\alpha}\mid L_{1,\beta}=y^{-\beta}$ is equivalent to that of $\tilde{\textbf{V}}=\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V})\mid L_{1,V}=y^{-\beta}$ , with distribution denoted $\mathrm{PD}_{\alpha|\beta}(\alpha\mid y)\,:\!=\,\mathrm{PK}_{\alpha}\big(\omega^{(y)}_{{\beta}/{\alpha},\beta}\cdot f_{\alpha}\big)$ as in (3.6), where $\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)$ is a ratio of stable densities and hence does not have an explicit form for general $0<\beta<\alpha<1$ . We now present results for the EPPF of the $\mathrm{PD}_{\alpha|\beta}(\alpha\mid y)$ Gibbs partition of [n]. We first note that since $T_{\alpha}\mid K^{[\alpha]}_{n}=k$ is equivalent in distribution to $Y^{n-k\alpha}_{\alpha,k\alpha}$ with density $f^{(n-k\alpha)}_{\alpha,k\alpha}$ , the EPPF can be expressed as $\big[\int_{0}^{\infty}\omega^{(y)}_{{\beta}/{\alpha},\beta}(s)f^{n-k\alpha}_{\alpha,k\alpha}(s)\,{\mathrm{d}} s\big] p_{\alpha}(n_{1},\ldots,n_{k})$ , where the first integral term is the density of $Y^{(n-k\alpha)}_{\alpha,k\alpha}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}}$ divided by $f_{\beta}(y)$ , and does not have an obvious recognizable form. However, we can use the approach in [Reference Ho, James and Lau20] to express $\omega^{(y)}_{{\beta}/{\alpha},\beta}$ in terms of Fox-H functions [Reference Mathai, Saxena and Haubold29], leading to the expression for the EPPF in terms of Fox-H functions given in Appendix 10.

The next result provides a more revealing expression that is not obvious.

Theorem 5.1. The EPPF of the $\mathrm{PD}_{\alpha|\beta}(\alpha\mid y)$ Gibbs partition of [n], corresponding to sampling n times from mass partitions with distribution described in (5.3), can be expressed as

(5.4) \begin{equation} p_{\alpha|\beta}(n_{1},\ldots,n_{k}\mid y) \,:\!=\, \Bigg[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(j)\frac{f^{(n-j\beta)}_{\beta,j\beta}(y)}{f_{\beta}(y)}\Bigg] p_{\alpha}(n_{1},\ldots,n_{k}), \end{equation}

where $\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(j)=\mathbb{P}_{{\beta}/{\alpha},0}(K_{k}=j)$ is the distribution of the number of blocks in a $\mathrm{PD}({\beta}/{\alpha},0)$ partition of [k], and $\sum_{j=1}^{k}\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(j){f^{(n-j\beta)}_{\beta,j\beta}(y)}$ is the conditional density of $T_{\beta}\mid K^{[\alpha]}_{n}=k$ for the number of blocks $K^{[\alpha]}_{n}$ in a $\mathrm{PD}(\alpha,0)$ partition of [n], with $T_{\beta}\,:\!=\,T_{\alpha}(T_{{\beta}/{\alpha}}(1)) \overset{{\mathrm{d}}}= T_{\alpha}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}}$ being equivalent to the inverse local time at 1 of $\textbf{V}_{\beta}\sim \mathrm{PD}(\beta,0)$ .

Proof. The expression for the EPPF is the conditional distribution of a $\mathrm{PD}(\alpha,0)$ partition of [n] given $T_{\beta}=y$ . The joint distribution may be expressed as in (5.4) in terms of the marginal EPPF $p_{\alpha}(n_{1},\ldots,n_{k})$ and the conditional density of $T_{\beta}\mid K^{[\alpha]}_{n}=k$ . It remains to show that $T_{\beta}\mid K^{[\alpha]}_{n}=k$ agrees with the expression in (5.4) as indicated. Recall that $L_{1,\beta}=L_{{\beta}/{\alpha}}(L_{1,\alpha})$ and hence the corresponding inverse local time at 1 is $T_{\beta}\,:\!=\,T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1))$ , corresponding to the coagulation operation dictated by $F_{\beta}=F_{\alpha}\circ F_{{\beta}/{\alpha}}$ , as expressed in (3.3). Sampling from $F_{\alpha}\circ F_{{\beta}/{\alpha}}$ , that is, according to variables $\big(F^{-1}_{{\beta}/{\alpha}}(F^{-1}_{\alpha}(U'_{i})), i\in [n]\big)$ , where $F^{-1}$ denotes the quantile function, it follows that this procedure produces a $\mathrm{PD}(\beta,0)$ partition of [n] with $K^{[\beta]}_{n}\overset{{\mathrm{d}}}=K^{[{\beta}/{\alpha}]}_{K^{[\alpha]}_{n}}$ blocks, where the two components are independent. Furthermore, the order matters, giving $K^{[\alpha]}_{n}$ the interpretation as the number of blocks to be merged, according to a $\mathrm{PD}({\beta}/{\alpha},0)$ partition of [k], for $K^{[\alpha]}_{n}=k\leq n$ . Now, from [Reference Ho, James and Lau21], $T_{\beta}\overset{{\mathrm{d}}}=Y^{n-K^{[\beta]}_{n}\beta}_{\beta,K^{[\beta]}_{n}\beta}$ . Hence $T_{\beta}\mid K^{[\alpha]}_{n}=k$ is equivalent to $Y^{(n-K^{[{\beta}/{\alpha}]}_{k}\beta)}_{\beta,K^{[{\beta}/{\alpha}]}_{k}\beta}$ , which, using (5.1), leads to the description of the density of $T_{\beta}\mid K^{[\alpha]}_{n}=k$ appearing in (5.4).

Remark 5.1. The previous result is equivalent to showing that $Y^{(n-K^{[{\beta}/{\alpha}]}_{k}\beta)}_{\beta,K^{[{\beta}/{\alpha}]}_{k}\beta}\overset{{\mathrm{d}}} = Y^{(n-k\alpha)}_{\alpha,k\alpha)}\times T^{{1}/{\alpha}}_{{\beta}/{\alpha}}$ , which can be deduced directly using the subordinator representation [Reference Ho, James and Lau21, Theorem 2.1 and Proposition 2.1] and decompositions of beta variables.

5.2. EPPF of $\mathrm{FRAG}_{\alpha,-\beta}(V) \sim \mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$

Recall from [Reference Gnedin and Pitman15, Reference Pitman32] (see also [Reference Ho, James and Lau21]) that if $\textbf{V}\sim \mathrm{PK}_{\beta}(h\cdot f_{\beta})$ with $\mathbb{E}[h(T_{\beta})]=1$ , then the EPPF of its associated Gibbs partition of [n] is described as $p^{[\gamma]}_{\beta}(n_{1},\ldots,n_{k})={\Psi}^{[\beta]}_{n,k} \times p_{\beta}(n_{1},\ldots,n_{k})$ , where $\Psi^{[\beta]}_{n,k}=\mathbb{E}_{\beta}\big[h(T_{\beta})\mid K^{[\beta]}_{n}=k\big]$ and, for clarity, $K^{[\beta]}_{n}$ is the number of blocks of a $\mathrm{PD}(\beta,0)$ partition of [n].

Theorem 5.1 leads to the EPPF corresponding to $\tilde{\textbf{V}}=\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V})\sim \mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ , or any variable in $\mathcal{P}_{\infty}$ having the same distribution.

Proposition 5.1. Suppose that, for $0<\beta<\alpha<1$ , $\tilde{\textbf{V}} \sim \mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ , where $\tilde{h}_{{\beta}/{\alpha}}(v)\,:\!=\,\mathbb{E}_{{\beta}/{\alpha}}\big[h\big(vT^{{1}/{\alpha}}_{{\beta}/{\alpha}}\big)\big]$ . Then, the $\mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ EPPF of the associated Gibbs partition of [n] can be expressed as

(5.5) \begin{equation} \Bigg[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(j)\Psi^{[\beta]}_{n,j}\Bigg]p_{\alpha}(n_{1},\ldots,n_{k}), \end{equation}

and we have the identity, for $T_{\beta}\,:\!=\,T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1))$ ,

\begin{align*} \mathbb{E}_{\alpha}\big[\tilde{h}_{{\beta}/{\alpha}}(T_{\alpha})\mid K^{[\alpha]}_{n}=k\big] = \mathbb{E}\big[h(T_{\beta})\mid K^{[\alpha]}_{n}=k\big] = \sum_{j=1}^{k}\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(j)\Psi^{[\beta]}_{n,j}. \end{align*}

Proof. The EPPF is equivalent to $\int_{0}^{\infty}p_{\alpha|\beta}(n_{1},\ldots,n_{k}\mid y)h(y)f_{\beta}(y)\,{\mathrm{d}} y$ , and hence the result follows from (5.4) in Theorem 5.1.

Remark 5.2. The expression in (5.5) provides a description of any mass partition with distribution $\mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ where $\tilde{h}_{{\beta}/{\alpha}}(v)\,:\!=\,\mathbb{E}_{{\beta}/{\alpha}}\big[h\big(vT^{{1}/{\alpha}}_{{\beta}/{\alpha}}\big)\big]$ , regardless of whether or not it actually arises from a fragmentation operation.

As a check, in the case where $\tilde{\textbf{V}}\overset{{\mathrm{d}}}=\textbf{V}_{\alpha}\sim \mathrm{PD}(\alpha,\theta)$ and hence $\textbf{V}\overset{{\mathrm{d}}}=\textbf{V}_{\beta}\sim \mathrm{PD}(\beta,\theta)$ , (5.5) must satisfy

\begin{equation*} \sum_{j=1}^{k}\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(j) \frac{\Gamma({\theta}/{\beta}+j)}{\Gamma({\theta}/{\beta}+1)\Gamma(j)} = \frac{\Gamma({\theta}/{\alpha}+k)}{\Gamma({\theta}/{\alpha}+1)\Gamma(k)}.\end{equation*}

However, this is satisfied since it agrees with [Reference Pitman33, Exercise 3.2.9, p. 66], with k in place of n. We have the following corollary in the case of ${\beta}/{\alpha}=\frac12$ .

Corollary 5.1. Specializing Theorem 5.1 to the case of ${\beta}/{\alpha}=\frac12$ , where $\textbf{V}\sim \mathrm{PK}_{\alpha/2}(h\cdot f_{{\alpha}/{2}})$ and $\Psi^{[{\alpha}/{2}]}_{n,j}=\mathbb{E}_{\alpha/2}\big[h(T_{\alpha/2})\mid K^{[{\alpha}/{2}]}_{n} = j\big]$ , the $\mathrm{PK}_{\alpha}(\tilde{h}_{{1}/{2}}\cdot f_{\alpha})$ EPPF in (5.5) becomes

\begin{equation*} \Bigg[\sum_{j=1}^{k}{\binom{2k-j-1}{k-1}}2^{j+1-2k}\Psi^{[{\alpha}/{2}]}_{n,j}\Bigg]p_{\alpha}(n_{1},\ldots,n_{k}). \end{equation*}

5.3. $\mathbf{PD}\big(\alpha,-\frac12\big)$ fragmentation of a Brownian excursion partition conditioned on its local time

Following [Reference Pitman32, Section 8] and [Reference Pitman33, Section 4.5, p. 90], let $(P_{\ell,0})_{\ell\ge 1}\sim \mathrm{PD}\big(\frac12,0\big)$ denote the ranked excursion lengths of a standard Brownian motion $B\,:\!=\,(B_{t}\colon t\in [0,1])$ , with corresponding local time at 0 up to time 1 given by $\tilde{L}_{1}\overset{{\mathrm{d}}}=(2T_{1/2})^{-1/2} \overset{{\mathrm{d}}}=|B_{1}|$ . Then, it follows that $(P_{\ell,0})_{\ell\ge 1}\mid \tilde{L}_{1}=s$ has a $\mathrm{PD}\big(\frac{1}{2}\mid\frac{1}{2}s^{-2}\big)$ distribution. Furthermore, with respect to $(P_{\ell}(s))_{\ell\ge 1}\sim\mathrm{PD}\big(\frac{1}{2}\mid\frac{1}{2}s^{-2}\big)$ , we describe the special $\beta=\frac12$ explicit case of the Gibbs partitions (EPPF) of [n] in terms of Hermite functions as derived in [Reference Pitman32] (see also [Reference Pitman33, Section 4.5]) as

(5.6) \begin{equation} p_{1/2}\big(n_{1},\ldots,n_{k}\mid\tfrac{1}{2}s^{-2}\big) = s^{k-1}\tilde{H}_{k+1-2n}(s)\frac{\Gamma(n)}{2^{1-n}\Gamma(k)}p_{\frac{1}{2}}(n_{1},\ldots,n_{k}),\end{equation}

where, for U(a,b,c) a confluent hypergeometric function of the second kind (see [Reference Lebedev27, p. 263]),

\begin{align*}\tilde{H}_{-2q}(s) = 2^{-q}U\bigg(q,\frac{1}{2}, \frac{s^2}{2}\bigg) =\sum_{\ell=0}^{\infty}\frac{{(-s)}^{\ell}}{\ell!}\frac{\Gamma(q+{\ell}/{2})}{2\Gamma(2q)}2^{q+{\ell}/{2}}\end{align*}

is a Hermite function of index $-2q$ . That is, in (5.2), $\mathbb{G}^{(n,k)}_{1/2}\big(\frac{1}{2}s^{-2}\big) = 2^{n-k}s^{k-1}\tilde{H}_{k+1-2n}(s)$ .

Proposition 5.2. Suppose that $\mathbf{P}_{1/2}(s) \,:\!=\, (P_{\ell}(s))_{\ell\ge 1} \sim \mathrm{PD}\big(\frac{1}{2}\mid\frac{1}{2}s^{-2}\big)$ . Then, for $\alpha>\frac12$ , $\mathbf{P}_{\alpha|{1}/{2}}(s) \,:\!=\, \mathrm{FRAG}_{\alpha,-{1}/{2}}(\mathbf{P}_{1/2}(s)) \sim \mathrm{PD}_{\alpha|{1}/{2}}\big(\alpha\mid\frac{1}{2}s^{-2}\big)$ , with corresponding EPPF expressed in terms of a mixture of Hermite functions,

(5.7) \begin{equation} \Bigg[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{{1}/{2\alpha},0}(j)2^{n-1}s^{j-1}\tilde{H}_{j+1-2n}(s) \frac{\Gamma(n)}{\Gamma(j)}\Bigg]p_{\alpha}(n_{1},\ldots,n_{k}). \end{equation}

Proof. The result follows as a special case of Theorems 3.1 and 5.1, and otherwise using the explicit form of the EPPF in (5.6).

Remark 5.3. In order to obtain a partition of [n] corresponding to the EPPF in (5.7), we can sample from (5.6) via the prediction rules indicated in [Reference Pitman32, (111) and (112)], and then employ the $\mathrm{CRP}\big(\alpha,-\frac12\big)$ fragmentation scheme in (3.2).

5.4. $\mathbf{FRAG}_{\alpha,-\beta}$ for the Mittag-Leffler class

We now present results for an application of the $\mathrm{FRAG}_{\alpha,-\beta}$ operator to the most basic case of the Mittag-Leffler class as described in [Reference Ho, James and Lau21] (see also [Reference James23]). Consider again $\mathbf{V}_{\beta}\sim \mathrm{PD}(\beta,0)$ . Recall that for $\lambda>0$ , the Laplace transform of $L_{1,\beta}\overset{{\mathrm{d}}}=T^{-\beta}_{\beta}$ equates to the Mittag-Leffler function (see, for instance, [Reference Gorenflo, Kilbas, Mainardi and Rogosin16]), expressed as

\begin{align*}\mathrm{E}_{\beta,1}(-\lambda) = \mathbb{E}\big[{\text e}^{-\lambda T^{-\beta}_{\beta}}\big] =\sum_{\ell=0}^{\infty}\frac{{(-\lambda)}^{\ell}}{\Gamma(\beta\ell+1)}.\end{align*}

Independent of $\mathbf{V}_{\beta}$ , let $(N(s),s\ge 0)$ denote a standard Poisson process, where $\mathbb{E}[N(s)]=s$ , and consider the mixed Poisson process $(N(t L_{1,\beta}), t\ge 0)$ . Then, as shown in [Reference Ho, James and Lau21], $\textbf{V}_{\beta}\mid N(\lambda L_{1,\beta})=0$ has a $\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ distribution of the form

(5.8) \begin{equation} \int_{0}^{\infty}\mathrm{PD}(\beta\mid t) \frac{{\text e}^{-\lambda t^{-\beta}}}{\mathrm{E}_{\beta,1}(-\lambda)}f_{\beta}(t)\,{\mathrm{d}} t,\end{equation}

where $h(t)={\text e}^{-\lambda t^{-\beta}}/\mathrm{E}_{\beta,1}(-\lambda)$ . Furthermore, from [Reference Ho, James and Lau21, Proposition 4.5] its corresponding EPPF for a partition of [n] can be expressed as

\begin{align*}\frac{\mathrm{E}^{(k)}_{\beta,n}(-\lambda)}{\mathrm{E}_{\beta,1}(-\lambda)}p_{\beta}(n_1,\ldots,n_{k}),\end{align*}

where, as in [Reference Ho, James and Lau21, Proposition 4.2],

\begin{equation*} \mathrm{E}^{(k)}_{\beta,n}(-\lambda) = \sum_{\ell=0}^{\infty}\frac{(\!-\lambda)^{\ell}}{\ell!}\frac{\Gamma(k+\ell)\Gamma(n)}{\Gamma(k)\Gamma(\beta\ell + n)}.\end{equation*}

We now describe properties of the fragmented process; additional details to verify the calculations are provided within the statement of the result.

Proposition 5.3. Let $\textbf{V}$ have distribution specified in (5.8), and otherwise consider the setting in Theorem 3.1.

1. (i) $\tilde{\textbf{V}}=\mathrm{FRAG}_{\alpha,-\beta}(\textbf{V}) \sim \mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot f_{\alpha})$ , where

   \begin{align*} \tilde{h}_{{\beta}/{\alpha}}(s) = \frac{\mathrm{E}_{{\beta}/{\alpha},1}(-\lambda s^{-\beta})}{\mathrm{E}_{\beta,1}(-\lambda)} = \frac{\int_{0}^{\infty}{\mathrm{e}}^{-\lambda s^{-\beta}y^{-{\beta}/{\alpha}}}f_{{\beta}/{\alpha}}(y)\,{\mathrm{d}} y} {\mathrm{E}_{\beta,1}(-\lambda)}. \end{align*}

2. (ii) Its EPPF for a partition of [n] can be expressed as

   \begin{align*} \frac{\sum_{\ell=0}^{\infty}\frac{{(-\lambda)}^{\ell}}{\Gamma({\beta\ell}/{\alpha}+1)} \frac{\Gamma(n)\Gamma({\beta\ell}/{\alpha}+k)}{\Gamma(k)\Gamma(\beta\ell+n)}} {\mathrm{E}_{\beta,1}(-\lambda)}p_{\alpha}(n_1,\ldots,n_{k}), \end{align*}
   which follows from the use of $\mathbb{E}[T^{-\beta\ell}_{\alpha}\mid K_{n}=k]$ in (1.2).

3. (iii) From (5.5) in Proposition 5.1, we have the identity

   \begin{align*} \sum_{j=1}^{k}\mathbb{P}^{(k)}_{{\beta}/{\alpha},0}(\,j)\mathrm{E}^{(\,j)}_{\beta,n}(-\lambda) = \sum_{\ell=0}^{\infty}\frac{{(-\lambda)}^{\ell}}{\Gamma({\beta\ell}/{\alpha}+1)} \frac{\Gamma(n)\Gamma({\beta\ell}/{\alpha}+k)}{\Gamma(k)\Gamma(\beta\ell+n)}. \end{align*}

6. Coagulation and fragmentation of generalized gamma models

For any $0 < \beta < 1$ , let $(\tau_{\beta}(s), s \ge 0)$ denote a generalized gamma subordinator specified by its Laplace transform: $\mathbb{E}[{\text e}^{-w\tau_{\beta}(s)}] = {\text e}^{-s[(1+w)^{\beta}-1]}$ . The generalized gamma subordinator, along with corresponding mass partitions and bridges defined by normalization, say $({\tau_{\beta}(\zeta u)}/{\tau_{\beta}(\zeta)}\colon u \in [0,1])$ for $\zeta>0$ , as described in [Reference Pitman32], arises in numerous contexts. However, for the purpose of this exposition, the reader may refer to its role in the construction of $\mathrm{PD}(\beta, \theta)$ distributions for $\theta>0$ , as described in [Reference Pitman and Yor37, Proposition 21].

Based on notes provided by Jim Pitman [Reference Pitman34], with relevance to species sampling and latent feature models, we can relate this and a more general size-biased class to $\mathbf{V}_{\beta} \sim \mathrm{PD}(\beta, 0)$ as follows. Again, let $(N(s)\colon s \ge 0)$ denote a standard Poisson process independent of $\mathbf{V}_{\beta}$ . Then the distribution of $\textbf{V}_{\beta} \mid N(\zeta^{{1}/{\beta}} T_{\beta}(1)) = m$ corresponds to the size-biased general gamma $\mathbf{V} \,:\!=\, (V_{k})_{k \ge 1} \sim \mathbb{P}^{[m]}_{\beta}(\zeta) \,:\!=\, \mathrm{PK}_{\beta}\big(r^{[m]}_{\beta,\zeta} \cdot f_{\beta}\big)$ , where $r^{[m]}_{\beta,\zeta}(t) = {t^{m} {\text e}^{-\zeta^{{1}/{\beta}} t}}/{\mathbb{E}\big[T^{m}_{\beta} {\text e}^{-\zeta^{{1}/{\beta}} T_{\beta}}\big]}$ for $m = 0, 1, 2, \ldots$ , as described in [Reference Ho, James and Lau21, Reference James24, Reference James25].

The case of $m = 0$ corresponds to the well-known case of the distribution of the mass partitions of $({\tau_{\beta}(\zeta u)}/{\tau_{\beta}(\zeta)}\colon u \in [0, 1])$ . The other cases are less studied but arise in various contexts. Here, for brevity, we show how to use Proposition 4.1 to easily identify laws and explicit constructions of (dependent) random measures leading to a coagulation/fragmentation duality in the case of $m = 1$ . That is, suppose that $\mathbf{V} \sim \mathbb{P}^{[1]}_{\beta}(\zeta) \,:\!=\, \mathrm{PK}_{\beta}\big(r^{[1]}_{\beta,\zeta} \cdot f_{\beta}\big)$ , where $r^{[1]}_{\beta,\zeta}(t) = {\zeta^{{1}/{\beta}-1} t {\text e}^{-\zeta^{{1}/{\beta}} t} {\text e}^{\zeta}}/{\beta}$ .

The case of $m=1$ also allows us to recover the Poisson–Dirichlet coagulation/fragmentation duality results of [Reference Pitman31] based on independent $\mathrm{PD}(\alpha,\theta)$ and $\mathrm{PD}({\beta}/{\alpha},{\theta}/{\alpha})$ distributions for the general case of $\theta>-\beta$ . The case of $m=0$ is fairly straightforward but we can recover the duality only for $\theta>0$ using [Reference Pitman and Yor37, Proposition 21]. Results for general m using Proposition 4.1 are also manageable but require too many additional details for the present exposition.

Let $\big(\tau^{(1)}_{\beta}(\zeta v), v \in [0,1]\big)$ denote the subordinator $\tau_{\beta}$ run up to a length of $\zeta$ , and let $\tau^{(2)}_{\beta}(G_{({1-\beta})/{\beta}}) \overset{{\mathrm{d}}}= G_{1-\beta}$ denote its total mass over the disjoint interval of length $G_{({1-\beta})/{\beta}}$ , where $G_{a}\sim \mathrm{Gamma}(a,1)$ for $a>0$ (see [Reference Pitman and Yor37, Proposition 21]. This implies that $\tau_{\beta}(\zeta+G_{({1-\beta})/{\beta}})\overset{{\mathrm{d}}}= \tau^{(1)}_{\beta}(\zeta) + \tau^{(2)}_{\beta}(G_{({1-\beta})/{\beta}})$ . Although not well known, it follows from [Reference James24] that the corresponding random distribution function (bridge) in the case of $\mathbf{V}\sim\mathbb{P}^{[1]}_{\beta}(\zeta)$ has the representation, for $v\in[0,1]$ ,

(6.1) \begin{equation} F_{\mathbf{V}}(v) \overset{{\mathrm{d}}}{=} \frac{\tau_{\beta}(\zeta v+G_{({1-\beta})/{\beta}}1_{\{U_{1}\leq v\}})}{\tau_{\beta}(\zeta+G_{({1-\beta})/{\beta}})}, \quad {\text {with }} T_{\mathbf{V}}\overset{{\mathrm{d}}} = \frac{\tau_{\beta}(\zeta+G_{({1-\beta})/{\beta}})}{\zeta^{{1}/{\beta}}}.\end{equation}

In this representation, we consider $U_{1}$ to be the first atom picked in a sample. This is the concomitant of the mass corresponding to the first size-biased pick, say $\tilde{P}_{1,\beta}(\zeta)$ , from $\mathbf{V}$ appearing in the size-biased representation

(6.2) \begin{equation} F_{\mathbf{V}}(v) \,:\!=\, (1 - \tilde{P}_{1,\beta}(\zeta)) \frac{\tau^{(1)}_{\beta}(\zeta v)}{\tau^{(1)}_{\beta}(\zeta)} + \tilde{P}_{1,\beta}(\zeta) 1_{\{U_{1} \leq v\}},\end{equation}

where $\tilde{P}_{1,\beta}(\zeta) \,:\!=\, {\tau^{(2)}_{\beta}(G_{({1 - \beta})/{\beta}})}/{\tau_{\beta}(\zeta + G_{({1 - \beta})/{\beta}})}$ and we use $\tau_{\beta}(\zeta + G_{({1 - \beta})/{\beta}}) \,:\!=\, \tau^{(1)}_{\beta}(\zeta) + \tau^{(2)}_{\beta}(\gamma_{({1 - \beta})/{\beta}})$ . Our purpose in the next result is to determine, given that $\mathbf{V} \sim \mathbb{P}^{[1]}_{\beta}(\zeta) \,:\!=\, \mathrm{PK}_{\beta}\big(r^{[1]}_{\beta,\zeta} \cdot f_{\beta}\big)$ , explicit equality in distribution constructions of $(F_{\tilde{\mathbf{V}}}, F_{\mathbf{Q}})$ satisfying $F_{\textbf{V}}=F_{\tilde{\textbf{V}}}\circ F_{\mathbf{Q}}$ , where $\tilde{\textbf{V}}\overset{{\mathrm{d}}}=\mathrm{FRAG}_{\alpha, -\beta}(\textbf{V})$ and $\mathbf{Q}$ is the coagulating mass partition. Furthermore, we show how these can be used to recover the duality in the Poisson–Dirichlet case of [Reference Pitman31] for all $\theta > -\beta$ .

Here, we apply Proposition 4.1, as well as distribution theory connected with these generalized gamma models and the Poisson–Dirichlet family, to identify all the relevant distributions in the next result, which is new.

Proof. The results follow from an application of Proposition 4.1 using $h(t)=r^{[1]}_{\beta,\zeta}(t)$ , the distributional representation of $T_{\mathbf{Q}}$ , and the appropriate Gamma randomization to obtain independent PD laws. As mentioned, the generalized gamma subordinator representation of $T_{\mathbf{Q}}$ , $T_{\mathbf{V}}$ , and Poisson–Dirichlet distributional identities can be found in [Reference Ho, James and Lau21, Reference James24, Reference James25]. For more clarity it is straightforward to show that $r^{[1]}_{{\beta}/{\alpha},\zeta}(y)=\int_{0}^{\infty}r^{[1]}_{\beta,\zeta}(sy^{{1}/{\alpha}})f_{\alpha}(s)\,{\mathrm{d}} s$ . As further checks, in verifying (iii) and (iv) we have

\begin{align*} \mathbb{E}\Big[r^{[1]}_{\beta,G_{({\theta+\beta})/{\beta}}}(t)\Big] = \frac{\int_{0}^{\infty}r^{[1]}_{\beta, x}(t)x^{(({\theta+\beta})/{\beta})-1}{\text e}^{-x}\,{\mathrm{d}} x} {\Gamma({\theta}/{\beta}+1)} = \frac{t^{-\theta}}{\mathbb{E}[T^{-\theta}_{\beta}]}. \end{align*}

Set $t = s y^{{1}/{\alpha}}$ to obtain the independence.

Although $F_{\mathbf{V}}(v) = F_{\tilde{\mathbf{V}}}(F_{\mathbf{Q}}(v))$ is a condition of the result, we can directly check that the composition of the subordinator representations in (6.3) and (6.4) results in a $\mathbb{P}^{[1]}_{\beta}(\zeta)\,:\!=\, \mathrm{PK}_{\beta}\big(r^{[1]}_{\beta,\zeta}\cdot f_{\beta}\big)$ -bridge. A key point is that, by definition, as the concomitant of the first size-biased pick, $U_{1} = F^{-1}_{\mathbf{Q}}(\tilde{U}_{1})$ , where $F^{-1}_{\mathbf{Q}}$ denotes the quantile function of $F_{\mathbf{Q}}$ . This results in $\tau_{\alpha}\big(\tau_{{\beta}/{\alpha}}(\zeta + G_{({\alpha - \beta})/{\beta}})F_{\mathbf{Q}}(v) +G_{({1 - \alpha})/{\alpha}}1_{\{\tilde{U}_{1} \leq F_{\mathbf{Q}}(v)\}}\big)$ , written as

\begin{align*} \tau_{\alpha}\big(\tau_{{\beta}/{\alpha}}\big(\zeta v + G_{({\alpha - \beta})/{\beta}}1_{\{U_{1} \leq v\}}\big) + G_{({1 - \alpha})/{\alpha}}1_{\{U_{1} \leq v\}}\big) \overset{{\mathrm{d}}}{=} \tau_{\beta}\big(\zeta v + G_{({1 - \beta})/{\beta}}1_{\{U_{1} \leq v\}}\big),\end{align*}

where the term on the right-hand side involves using the facts discussed around the construction of (6.2) and (6.3), where we can take $\tau_{{\beta}/{\alpha}}(G_{({\alpha-\beta})/{\beta}})\overset{{\mathrm{d}}}=G_{({\alpha-\beta})/{\alpha}}$ , $\tau_{\alpha}(G_{({\alpha-\beta})/{\alpha}}+G_{({1-\alpha})/{\alpha}})\overset{{\mathrm{d}}}=G_{1-\beta}\overset{{\mathrm{d}}}=\tau_{\beta}(G_{({1-\beta})/{\beta}})$ , and $\tau_{\alpha}(\tau_{{\beta}/{\alpha}}(\zeta v))\overset{{\mathrm{d}}}=\tau_{\beta}(\zeta v)$ . This leads to the representations in (6.1) and (6.2).

Remark 6.2. If $\textbf{V}_{\alpha}\sim \mathrm{PD}(\alpha,0)$ independent of $\textbf{Q}_{{\beta}/{\alpha}}\sim \mathrm{PD}({\beta}/{\alpha},0)$ , it is evident that

\begin{align*} (\textbf{V}_{\alpha},\textbf{Q}_{{\beta}/{\alpha}})\mid N(\zeta^{{1}/{\beta}}T_{\alpha}(T_{{\beta}/{\alpha}}(1))) = m \sim \mathrm{P}^{{\beta}/{\alpha}}_{\alpha}\big(r^{[m]}_{\beta,\zeta}\big). \end{align*}