# Inverse Clustering of Gibbs Partitions via independent fragmentation and
dual dependent coagulation operators

Man-Wai<EMAIL_ADDRESS>[ Lancelot F.
<EMAIL_ADDRESS>[ John W<EMAIL_ADDRESS>[ Department of Information Systems, Business Statistics and Operations
Management, The Hong Kong University of Science and Technologypresep=, ]e1,e2
Department of Mathematics and Statistics, University of Western
Australiapresep=, ]e3

###### Abstract

Gibbs partitions of the integers generated by stable subordinators of index
$\alpha\in(0,1)$ form remarkable classes of random partitions where in
principle much is known about their properties, including practically
effortless obtainment of otherwise complex asymptotic results potentially
relevant to applications in general combinatorial stochastic processes, random
tree/graph growth models and Bayesian statistics. This class includes the
well-known models based on the two-parameter Poisson-Dirichlet distribution
which forms the bulk of explicit applications. This work continues efforts to
provide interpretations for a larger classes of Gibbs partitions by embedding
important operations within this framework. Here we address the formidable
problem of extending the dual, infinite-block, coagulation/fragmentation
results of Pitman [41], where in terms of coagulation they are based on
independent two-parameter Poisson-Dirichlet distributions, to all such Gibbs
(stable Poisson-Kingman) models. Our results create nested families of Gibbs
partitions, and corresponding mass partitions, over any $0<\beta<\alpha<1.$ We
primarily focus on the fragmentation operations, which remain independent in
this setting, and corresponding remarkable calculations for Gibbs partitions
derived from that operation. We also present definitive results for the dual
coagulation operations, now based on our construction of dependent processes,
and demonstrate its relatively simple application in terms of Mittag-Leffler
and generalized gamma models. The latter demonstrates another approach to
recover the duality results in [41].

60C05, 60G09,

60G57,60E99,

Brownian and Bessel processes, coagulation/fragmentation duality, Gibbs
partitions, Poisson Dirichlet distributions, stable Poisson-Kingman
distributions,

###### keywords:

[class=AMS]

###### keywords:

, and

##  1 Introduction

Gibbs (random) partitions, as developed in [42, Theorem 8] and subsequently
[17, 43], say, $\\{A_{1},\ldots,A_{K_{n}}\\}$ of the integers
$[n]=\\{1,2,\ldots,n\\},$ where $K_{n}\leq n,$ form remarkable classes of
infinitely exchangeable random partitions [43, Section 2.2], whose
distribution is defined consistently for each
$n\in\mathbb{N}=\\{1,2,\ldots\\}$ as the restriction of an exchangeable
partition probability function (EPPF) on $\mathbb{N}$ that has the
distinguishing feature of having Gibbs (product) form for each $n.$ In
principle, the Gibbs form is attractive in terms of practical implementation,
and much is known about the properties of this class, including practically
effortless obtainment of otherwise complex asymptotic results potentially
relevant to applications in combinatorial stochastic processes, random
tree/graph growth models and Bayesian statistics. See, for example, [4, 6, 10,
13, 15, 19, 22, 23, 25, 33, 34, 38, 44, 47, 48].

The class is defined for each $\alpha\in(-\infty,1),$ here we shall focus on
the case where $\alpha\in(0,1),$ which may be generated by an $\alpha$-stable
subordinator in relation to the work [39, 46, 42, 43] on excursion lengths of
Bessel processes. In particular, let $\mathcal{B}:=(B_{t},t>0)$ denote a
strong Markov process on $\mathbb{R}$ whose normalized ranked lengths of
excursions,
$(P_{i})\in\mathcal{P}_{\infty}=\\{\mathbf{s}=(s_{1},s_{2},\ldots):s_{1}\geq
s_{2}\geq\cdots\geq 0{\mbox{ and }}\sum_{k=1}^{\infty}s_{k}=1\\},$ follow a
Poisson Dirichlet law with parameters $(\alpha,0)$, for $0<\alpha<1,$ as
discussed in Pitman and Yor [46]. Denote this law as $\mathrm{PD}(\alpha,0)$
on the space of mass partitions summing to one, $\mathcal{P}_{\infty}.$ Let
$(L_{t};t>0)$ denote its local time starting at $0,$ and let
$T_{\alpha}(\ell)=\inf\\{t:L_{t}>\ell\\},\ell\geq 0$, denote its inverse local
time. In this case, $\mathbf{T}_{\alpha}:=(T_{\alpha}(t),t\geq 0)$ is an
$\alpha$-stable subordinator. For each $s,$
$T_{\alpha}(s)\overset{d}{=}s^{{1}/{\alpha}}T_{\alpha}(1),$ where
$T_{\alpha}(1):=T_{\alpha}$ is the inverse local time at $1$ with density
$f_{\alpha}(t)$ and Laplace transform $\mathbb{E}[{\mbox{e}}^{-\lambda
T_{\alpha}}]={\mbox{e}}^{-\lambda^{\alpha}}.$ Due to the scaling identity (see
[45]),

$L_{1}\overset{d}{=}\frac{L_{t}}{t^{\alpha}}\overset{d}{=}\frac{s}{[{{T_{\alpha}(s)]}^{\alpha}}}\overset{d}{=}T^{-\alpha}_{\alpha},$
(1.1)

the local time up to time $1,$ $L_{1}\overset{d}{=}T^{-\alpha}_{\alpha},$
follows a Mittag-Leffler distribution with density
$g_{\alpha}(z):=f_{\alpha}(z^{-1/\alpha})z^{-{1}/\alpha-1}/\alpha,$ satisfying

$L_{1}:=\Gamma(1-\alpha)^{-1}\lim_{\epsilon\rightarrow
0}\epsilon^{\alpha}|\\{i:P_{i}\geq\epsilon\\}|~{}\mathrm{a.s.}$ (1.2)

The process $F_{\alpha,0}(y):=\sum_{k=1}^{\infty}P_{k}\mathbb{I}_{\\{U_{k}\leq
y\\}},$ for $(U_{k})\overset{iid}{\sim}\mathrm{Uniform}[0,1]$ independent of
$(P_{i})\sim\mathrm{PD}(\alpha,0)$, is called a $\mathrm{PD}(\alpha,0)$
exchangeable bridge, see [6] for this terminology, with the equivalence

$F_{\alpha,0}(y)\overset{d}{=}\frac{T_{\alpha}(y)}{T_{\alpha}(1)}.$

In turn, sampling
$X_{1},\ldots,X_{n}|F_{\alpha,0}\overset{iid}{\sim}F_{\alpha,0},$ that is,
$(X_{i}\overset{d}{=}F_{\alpha,0}^{-1}(U^{\prime}_{i});$
$U^{\prime}_{i}\overset{iid}{\sim}\mathrm{Uniform}[0,1],i\in[n])$, produces
$K_{n}=k$ unique values
$(Y_{1},\ldots,Y_{k})\overset{iid}{\sim}\mathrm{Uniform}[0,1]$ and a
$\mathrm{PD}(\alpha,0)$-partition of $[n],$ $\\{A_{1},\ldots,A_{k}\\},$ where
$A_{j}=\\{i:X_{i}=Y_{j}\\},$ with size $|A_{j}|=n_{j},$ for $j=1,\ldots,k,$
with an EPPF denoted by the $\mathrm{PD}(\alpha,0)-\mathrm{EPPF}$,

$p_{\alpha}(n_{1},\ldots,n_{k})=\frac{\alpha^{k-1}\Gamma(k)}{\Gamma(n)}\prod_{j=1}^{k}(1-\alpha)_{n_{j}-1},$
(1.3)

where, for any non-negative integer $x$,
$(x)_{n}=x(x+1)\cdots(x+n-1)={\Gamma(x+n)}/{\Gamma(x)}$ denotes the Pochhammer
symbol. From this, the probability of the number of blocks $K_{n}=k$ can be
expressed as
$\mathbb{P}_{\alpha,0}^{(n)}(k):=\mathbb{P}_{\alpha,0}(K_{n}=k)={\alpha^{k-1}\Gamma(k)}S_{\alpha}(n,k)/{\Gamma(n)}$,
where
$S_{\alpha}(n,k)=[{\alpha^{k}k!}]^{-1}\sum_{j=1}^{k}(-1)^{j}\binom{k}{j}(-j\alpha)_{n}$
denotes the generalized Stirling number of the second kind.

###  1.1 $\alpha$-stable Poisson-Kingman distributions and Gibbs partitions

Now conditioning $(P_{i})$ on $T_{\alpha}=t$ (or $L_{1}=t^{-\alpha}$) leads to
the distribution of $(P_{i})|T_{\alpha}=t\sim\mathrm{PD}(\alpha|t)$, and for
$h(t)$ a non-negative function with $\mathbb{E}[h(T_{\alpha})]=1,$ one may, as
in [42], define the $\alpha$-stable Poisson-Kingman distribution with mixing
distribution $\nu(dt)/dt=h(t)f_{\alpha}(t),$ and write
$(P_{\ell})\sim\mathrm{PK}_{\alpha}(h\cdot f_{\alpha})$ defined as

$\mathrm{PK}_{\alpha}(h\cdot
f_{\alpha}):=\int_{0}^{\infty}\mathrm{PD}(\alpha|t)h(t)f_{\alpha}(t)dt=\int_{0}^{\infty}\mathrm{PD}(\alpha|s^{-\frac{1}{\alpha}})h(s^{-\frac{1}{\alpha}})g_{\alpha}(s)ds.$

Setting $h(t)=t^{-\theta}/\mathbb{E}[T^{-\theta}_{\alpha}]$, for
$\theta>-\alpha$, leads to $(P_{\ell})\sim\mathrm{PD}(\alpha,\theta),$
corresponding to the important two-parameter Poisson-Dirichlet distribution as
described in [39, 40, 42, 43, 46], whose size biased re-arrangement, say,
$(\tilde{P}_{\ell})\sim\mathrm{GEM}(\alpha,\theta),$ where
$\mathrm{GEM}(\alpha,\theta)$ is the two-parameter Griffiths-Engen-McCloskey
distribution, and is widely used in applications [6, 26, 39, 40, 43]. The
inverse local time at $1$ of a process with lengths
$(P_{\ell})\sim\mathrm{PD}(\alpha,\theta),$ say, $T_{\alpha,\theta},$ has
density
$f_{\alpha,\theta}(t)=t^{-\theta}f_{\alpha}(t)/\mathbb{E}[T^{-\theta}_{\alpha}]$,
and the corresponding local time at $1$ or its $\alpha$-diversity,
$T^{-\alpha}_{\alpha,\theta}\sim\mathrm{ML}(\alpha,\theta),$ denoting that it
has a $(\alpha,\theta)$ generalized Mittag-Leffler distribution with density
$g_{\alpha,\theta}(s)=s^{\theta/\alpha}g_{\alpha}(s)/\mathbb{E}[T^{-\theta}_{\alpha}]$.
When $(P_{\ell})\sim\mathrm{PK}_{\alpha}(h\cdot f_{\alpha})$,
$F(y)=\sum_{k=1}^{\infty}P_{k}\mathbb{I}_{\\{U_{k}\leq y\\}}$ is its
corresponding bridge defined similarly as $F_{\alpha,0}$, and sampling $n$
variables from $F$, as in the $\mathrm{PD}(\alpha,0)$ case, leads to the
general class of $\alpha\in(0,1)$ Gibbs partitions of $[n]$, with an EPPF ,
denoted by $\mathrm{PK}_{\alpha}(h\cdot f_{\alpha})-\mathrm{EPPF},$ as in [17,
42],

$p^{[\nu]}_{\alpha}(n_{1},\ldots,n_{k})={\Psi}^{[\alpha]}_{n,k}\times
p_{\alpha}(n_{1},\ldots,n_{k}),$ (1.4)

where, using the interpretation of the expressions derived by [17, 42] in
[25],

${\Psi}^{[\alpha]}_{n,k}=\mathbb{E}[h(T_{\alpha})|K_{n}=k]=\mathbb{E}\bigg{[}h\big{(}Y^{(n-k\alpha)}_{\alpha,k\alpha}\big{)}\bigg{]}.$

In the first expectation, $T_{\alpha}|K_{n}=k$ is evaluated for the
$\mathrm{PD}(\alpha,0)$ case with $K_{n}\sim\mathbb{P}_{\alpha,0}^{(n)}(k).$
The second equality follows from the fact that such a conditional random
variable equates in distribution to a variable
$Y^{(n-k\alpha)}_{\alpha,k\alpha},$ with density
$f^{(n-k\alpha)}_{\alpha,k\alpha}(t),$ such that (pointwise), as in [25, eq.
(2.13), p. 323],

$Y^{(n-k\alpha)}_{\alpha,k\alpha}\overset{d}{=}\frac{T_{\alpha,k\alpha}}{B_{k\alpha,n-k\alpha}}=\frac{T_{\alpha,n}}{B^{\frac{1}{\alpha}}_{k,\frac{n}{\alpha}-k}},$
(1.5)

where variables in each ratio are independent, and throughout, $B_{a,b}$
denotes a $\mathrm{Beta}(a,b)$ random variable. It is noteworthy that (1.5)
indicates that Mittag-Leffler variables play a role in the general $\alpha$
class of Gibbs partitions.

###  1.2 Interpreting Gibbs partitions via infinite block fragmentation and
coagulation operations

While the general class of $\mathrm{PK}_{\alpha}(h\cdot f_{\alpha})$
distributions and the corresponding Gibbs partitions exhibit many desirable
properties, most choices of $h(t)$ do not have any particular interpretation.
The most notable exceptions are the important $\mathrm{PD}(\alpha,\theta)$
distributions which dominate the broad literature. There are some additional
examples, such as the generalized gamma and Mittag-Leffler classes. Of
particular interest to us are classes that correspond to nested families of
mass partitions, whose marginal distributions follow some explicit collection
of $\mathrm{PK}_{\alpha}$ distributions, and whose Markovian dynamics
(dependence structure) may be described by some operations on
$\mathcal{P}_{\infty},$ and hence equivalently on spaces of integer
partitions. The two most notable examples of such families are the nested
families of the form $((P^{(j)}_{\ell}),j=0,1,2\ldots)$ represented by the
Poisson-Dirichlet laws $(\mathrm{PD}(\alpha,\theta+j\alpha),j=0,1,\ldots),$
and the laws $(\mathrm{PD}(\alpha,\theta+j),j=0,1,\ldots).$ The first such
collection is related to (dual) operations of size-biased deletion and
insertion, as described in [46, Propositions 34 and 35], and the latter may be
constructed by a single block $\mathrm{PD}(\alpha,1-\alpha)$ fragmentation
operation, fragmenting successively the size-biased pick of the indicated
families, leading to, in the initial case of $\mathrm{PD}(\alpha,\theta),$ an
increase of $\theta$ to $\theta+1,$ and inversely by a dual coagulation
operation as described in [14]. It is notable that the operations of size-
biased deletion, as described in [39, 42, 43], and single-block fragmentation
can be applied in principle to general families taking values on
$\mathcal{P}_{\infty},$ whereas the latter (dual) operations involve the usage
of independent beta distributed variables, which are particular to the
$\mathrm{PD}(\alpha,\theta)$ distribution. Common descriptions, for extensions
of size-biased deletion involving general $\mathrm{PK}_{\alpha}(h\cdot
f_{\alpha}),$ can be deduced from [39, 42], whereas fragmentation by
$\mathrm{PD}(\alpha,1-\alpha)$ has been treated in [25, 31]. Results for
specific examples of $h(t),$ requiring more detailed analysis, are also
discussed in those and related works.

This work continues efforts to provide interpretations for larger classes of
Gibbs partitions by embedding important operations within this framework. Here
we address the formidable problem of extending the dual, infinite block,
coagulation/fragmentation results of Pitman [41], that is, for
$0<\beta<\alpha<1,$ a dual relationship between
$V_{\beta,\theta}\sim\mathrm{PD}(\beta,\theta)$ and
$V_{\alpha,\theta}\sim\mathrm{PD}(\alpha,\theta),$ with appropriate
coagulation and fragmentation operations indicated, as in [43, Section 5.5],
by the following diagram

$\mathrm{PD}(\alpha,\theta)\qquad\begin{array}[]{c}\mathrm{PD}(\beta/\alpha,\theta/\alpha)-\mathrm{COAG}\\\
\xrightarrow{\hskip 72.26999pt}\\\ \xleftarrow{\hskip 72.26999pt}\\\
\mathrm{PD}(\alpha,-\beta)-\mathrm{FRAG}\end{array}\qquad\mathrm{PD}(\beta,\theta)$
(1.6)

This creates nested families of mass/integer partitions having laws
$(\mathrm{PD}(\alpha,\theta),0<\alpha<1).$ Setting $\theta=0,$
$\alpha={\mbox{e}}^{-t}$ and $\beta={\mbox{e}}^{-(t+s)},$ the (homogeneous)
continuous-time Markov operator $(\mathrm{PD}(e^{-s},0)-\mathrm{COAG},s\geq
0)$ coincides with the semi-group of the Bolthausen-Sznitman coalescent [9],
as discussed in [6, 41, 43]. Briefly, following [6, 43], we describe the
$\mathrm{PD}(\alpha,-\beta)-\mathrm{FRAG}$ fragmentation operator, hereafter
denoted as $\mathrm{FRAG}_{\alpha,-\beta},$ and corresponding
$\mathrm{PD}(\beta/\alpha,\theta/\alpha)-\mathrm{COAG}$ coagulation operator.
For $(P_{i})\in\mathcal{P}_{\infty},$ the $\mathrm{FRAG}_{\alpha,-\beta}$
operator is defined as, for $\mathrm{Rank}$ denoting the ranked re-arrangement
of masses,

$\mathrm{FRAG}_{\alpha,-\beta}((P_{i})):=\mathrm{Rank}(P_{i}(\hat{Q}^{(i)}_{\ell}),i\geq
1)\in\mathcal{P}_{\infty},$ (1.7)

where, for each $i,$
$\hat{Q}^{(i)}:=(\hat{Q}^{(i)}_{\ell})\in\mathcal{P}_{\infty},$ the collection
$(\hat{Q}^{(i)},i\geq 1)$ are iid $\mathrm{PD}(\alpha,-\beta)$ mass
partitions, taken independent of the input $(P_{i}).$ There is also the
property
$\mathrm{FRAG}_{\alpha,-\sigma}\circ\mathrm{FRAG}_{\sigma,-\beta}=\mathrm{FRAG}_{\alpha,-\beta}$
for all $0<\beta<\sigma<\alpha<1.$ In terms of partitions
$\\{A_{1},\ldots,A_{k}\\}$, one is shattering each block of the partition,
say, $A_{j}$ with size $|A_{j}|,$ by an independent
$\mathrm{PD}(\alpha,-\beta)$ partition of $|A_{j}|$ elements. We now proceed
to first describe the coagulation for more general independent laws on
$\mathcal{P}_{\infty},$ by using the equivalence in terms of compositions of
exchangeable bridges as in [43, Lemma 5.18] or [6]. Suppose that $F_{V}$ and
$G_{Q}$ are independent exchangeable bridges defined for $V$ and $Q$ in
$\mathcal{P}_{\infty},$ with respective distributions $\mathbb{P}_{V}$ and
$\mathbb{P}_{Q}.$ Then, for the input $\tilde{V},$
$V:=\mathbb{P}_{Q}-\mathrm{COAG}(\tilde{V})\in\mathcal{P}_{\infty}$ is
equivalent to the ranked re-arrangement of masses formed by the composition
$F_{V}(y)=F_{\tilde{V}}(G_{Q}(y)),$ for each $y\in[0,1],$ or simply
$F_{V}=F_{\tilde{V}}\circ G_{Q},$ where $V$ has distribution $\mathbb{P}_{V}.$
Letting $F_{\alpha,\theta}$ denote a $\mathrm{PD}(\alpha,\theta)$ bridge, the
$\mathrm{PD}({\beta}/{\alpha},{\theta}/{\alpha})-\mathrm{COAG}$ result in
[41], as indicated from left to right in (1.6), corresponds to
$F_{\beta,\theta}=F_{\alpha,\theta}\circ
G_{{\beta}/{\alpha},{\theta}/{\alpha}}.$

######  Remark 1.1.

Note $F_{\beta}:=F_{\beta,0}$ and the same is true for other variables with
$\theta=0.$

In this work, for $0<\beta<\alpha<1$ and general
$V\sim\mathrm{PK}_{\beta}(h\cdot f_{\beta}),$ where
$\mathbb{E}[h(T_{\beta})]=1,$ we shall apply the same independent
$\mathrm{FRAG}_{\alpha,-\beta}$ operator leading to explicit identification of
laws, and calculations for nested families of mass partitions and
corresponding Gibbs partitions over $\alpha\in(0,1).$ This represents the
primary focus of our work, however we also describe in detail how to construct
a natural dual coagulation operation via dependent compositions of bridges and
corresponding mass partitions, and demonstrate how easy it can be applied. We
note that, in the literature, coagulation operations of this sort are
generally defined for independent processes as in [43, Lemma 5.18]. While, in
general, fragmentation and coagulation operations are clearly defined and
hence straightforward to apply, the formidable challenge is to identify the
resulting relevant distributions, and find ones with tractability. The duality
result of [41], as described in (1.6), is achieved by working with the EPPF’s
of the corresponding random partitions of $[n].$ In principle one may try such
an approach to identify the various laws associated with
$V\sim\mathrm{PK}_{\beta}(h\cdot f_{\alpha}),$ or attempt via the explicit
constructions of exchangeable bridges. However, neither approach seems
feasible. Here working on the space of mass partitions, we show, in Sections 2
and 3, that the classes of (marginal) distributions
$\tilde{V}\sim\mathrm{PK}_{\alpha}(\tilde{h}_{{\beta}/{\alpha}}\cdot
f_{\alpha})$ and $Q\sim\mathrm{PK}_{\beta/\alpha}(\hat{h}_{\alpha}\cdot
f_{{\beta}/{\alpha}}),$ with

$\tilde{h}_{\frac{\beta}{\alpha}}(v):=\mathbb{E}_{\frac{\beta}{\alpha}}\left[h\bigg{(}vT^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}}\bigg{)}\right]\qquad{\mbox{
and
}}\qquad\hat{h}_{\alpha}(y):=\mathbb{E}_{\alpha}\left[h\big{(}T_{\alpha}y^{\frac{1}{\alpha}}\big{)}\right],$
(1.8)

may be interpreted as being equivalent in distribution to those arising from
$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V),$ and the marginal distribution of
the corresponding coagulator which leads to $F_{V}=F_{\tilde{V}}\circ G_{Q},$
respectively. In Sections 4.1-4.3, we obtain remarkable calculations for Gibbs
partitions, and related identities, derived from the
$\mathrm{FRAG}_{\alpha,-\beta}$ operation. Sections 4.4 and 4.5, provide
interesting identities, fixed point equations, and asymptotic results, which
has interpretations in relation to $D-$ and $T-\mathrm{partition}$ intervals
as discussed in [3, 43]. Sections 4.6 and 5 provide developments in relation
to the Mittag-Leffler class [25]. Section 6 demonstrates how to use our dual
coagulation and fragmentation operations to easily identify all the relevant
laws, and constructs duality results for generalized gamma models, and size
biased extensions. This presents another approach to recover the duality
results in [41] for all $\theta>-\beta.$

For nested models primarily related to the fragmentation operator see [5, 12].
In addition, [48], see also [16], applies the coag/frag duality on the space
of partitions of $[n]$ to $\infty$-gram natural language models. This
represents an application in Bayesian statistical machine learning involving
the usage of inverse clustering (via $\mathrm{FRAG}_{\alpha,-\beta}$
fragmentation) and merging (via $\mathrm{PD}(\alpha,\theta)$ coagulation) on
the space of partitions of $[n].$ Related to this, [32] constructs (nested)
hierarchical network/graph models using the coagulation fragmentation
operations in [41] and also [14]. For some other references on Gibbs-
partitions and $\alpha$-stable Poisson-Kingman models, see [4, 10, 13, 21, 24,
25, 36, 44]. See [22, 23] for other occurrences of the coag/frag operators in
the $\mathrm{PD}(\alpha,\theta)$ setting.

##  2 $\mathrm{PD}_{\alpha,-\beta}$ fragmentation of
$\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ mass partitions

In order to achieve our results, we work with independent stable subordinators
$\mathbf{T}_{\alpha}$ and $\mathbf{T}_{{\beta}/{\alpha}},$ and representations
of their relevant quantities under the independent $\mathrm{PD}(\alpha,0)$ and
$\mathrm{PD}({\beta}/{\alpha},0)$ distributions. The corresponding independent
local time processes are $(L_{\alpha}(t),t>0)$ and
$(L_{{\beta}/{\alpha}}(t),t\geq 0)$, satisfying (1.1) and (1.2), with local
times at $1,$ denoted respectively as
$L_{1,\alpha}\overset{d}{=}T^{-\alpha}_{\alpha}$ and
$L_{1,{\beta}/{\alpha}}\overset{d}{=}T^{-{\beta}/{\alpha}}_{{\beta}/{\alpha}}$,
playing the role of $L_{1}$, as we have described, and otherwise following the
more detailed description in [8], as it relates to the special case of the
duality [41, Theorem 14 and Corollary 15]. In particular,
$V_{\beta}\sim\mathrm{PD}(\beta,0)$ is formed by the independent coagulation,
$\mathrm{V}_{\beta}=\mathrm{PD}(\frac{\beta}{\alpha},0)-\mathrm{COAG}((V_{\alpha}))$,
equivalent to, as in [6, 7, 8, 43],

$\left(F_{\beta}(u)=F_{\alpha}\big{(}F_{\frac{\beta}{\alpha}}(u)\big{)},u\in[0,1]\right)\overset{d}{=}\left(\frac{T_{\alpha}\big{(}T_{\frac{\beta}{\alpha}}(u)\big{)}}{T_{\alpha}\big{(}T_{\frac{\beta}{\alpha}}(1)\big{)}},u\in[0,1]\right)$
(2.1)

and has local time at $1,$ $L_{1,\beta}=L_{{\beta}/{\alpha}}(L_{1,\alpha})$
and inverse local time at $1,$
$T_{\beta}:=T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1)).$ Conversely,
$V_{\alpha}=\mathrm{FRAG}_{\alpha,-\beta}(V_{\beta})\sim\mathrm{PD}(\alpha,0).$

######  Remark 2.1.

Note there is the well known distributional equivalence
$T_{\beta}\overset{d}{=}T_{\alpha}\times
T^{{1}/{\alpha}}_{{\beta}/{\alpha}}\overset{d}{=}T_{{\beta}/{\alpha}}\times
T^{{\alpha}/{\beta}}_{\alpha}.$ However, in the case of interpretation of the
$\mathrm{PD}({\beta}/{\alpha},0)$ coagulation, as in (2.1), the order matters
and thus we will only use $T_{\beta}\overset{d}{=}T_{\alpha}\times
T^{{1}/{\alpha}}_{{\beta}/{\alpha}}$.

Define

$\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)=\frac{\mathbb{P}\left({\big{[}L_{\frac{\beta}{\alpha}}(L_{1,\alpha})\big{]}}^{-\frac{1}{\beta}}\in
dy\left|L^{-\frac{1}{\alpha}}_{1,\alpha}=s\right.\right)}{\mathbb{P}\big{(}L^{-\frac{1}{\beta}}_{1,\beta}\in
dy\big{)}}=\frac{\alpha
y^{\alpha-1}f_{\frac{\beta}{\alpha}}\big{(}{(y/s)}^{\alpha}\big{)}}{s^{\alpha}f_{\beta}(y)}$
(2.2)

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

$\mathbb{P}\left(L^{-\frac{1}{\alpha}}_{1,\alpha}\in
ds\left|\big{[}L_{\frac{\beta}{\alpha}}(L_{1,\alpha})\big{]}^{-\frac{1}{\beta}}=y\right.\right)/ds=\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)f_{\alpha}(s),$
(2.3)

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

######  Theorem 2.1.

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

  1. (i)

$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V)\sim\mathrm{PK}_{\alpha}\bigg{(}\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\bigg{)}$ where

$\tilde{h}_{\frac{\beta}{\alpha}}(v):=\mathbb{E}_{\frac{\beta}{\alpha}}\left[h\bigg{(}vT^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}}\bigg{)}\right].$

That is, it has a local time at time $1,$ say, $\tilde{L}_{1,\tilde{V}}$, with
density
$\tilde{h}_{\frac{\beta}{\alpha}}(s^{-\frac{1}{\alpha}})g_{\alpha}(s)$.

  2. (ii)

$\tilde{V}|L_{1,V}=y^{-\beta}$ has the $\alpha$-stable Poisson-Kingman
distribution with, for each $y,$ and any $h(y),$ mixing density

$\mathbb{P}\left(\tilde{L}^{-\frac{1}{\alpha}}_{1,\tilde{V}}\in
ds\left|L^{-\frac{1}{\beta}}_{1,V}=y\right.\right)/ds=\mathbb{P}\left(L^{-\frac{1}{\alpha}}_{1,\alpha}\in
ds\left|\big{[}L_{\frac{\beta}{\alpha}}(L_{1,\alpha})\big{]}^{-\frac{1}{\beta}}=y\right.\right)/ds,$

as in (2.3).

  3. (iii)

The conditional distribution of $\tilde{V}|L^{-{1}/{\beta}}_{1,V}=y$ is
equivalent to the distribution of $V_{\alpha}|L^{-{1}/{\beta}}_{1,\beta}=y,$
which is

$\mathrm{PD}_{\alpha|\beta}(\alpha|y):=\int_{0}^{\infty}\mathrm{PD}(\alpha|s)\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)f_{\alpha}(s)ds=\mathrm{PK}_{\alpha}\left(\omega^{(y)}_{\frac{\beta}{\alpha},\beta}\cdot
f_{\alpha}\right),$ (2.4)

for $\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)$ defined in (2.2).

###### Proof.

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

$\mathbb{E}\left[\Omega\big{(}\mathrm{FRAG}_{\alpha,-\beta}(V)\big{)}\right]=\mathbb{\hat{E}}^{(\beta,0)}_{(\alpha,-\beta)}\left[\Omega\big{(}\mathrm{FRAG}_{\alpha,-\beta}(V_{\beta})\big{)}h\big{(}L^{-\frac{1}{\beta}}_{1,\beta}\big{)}\right].$
(2.5)

But, from [8, 41], as described in (1.6), this is equivalent to, for
$V_{\alpha}\sim\mathrm{PD}(\alpha,0),$

$\mathbb{E}\left[\Omega(V_{\alpha})h\bigg{(}{\big{[}L_{\frac{\beta}{\alpha}}(L_{1,\alpha})\big{]}}^{-\frac{1}{\beta}}\bigg{)}\right].$
(2.6)

It follows that
$V_{\alpha}|L^{-{1}/{\alpha}}_{1,\alpha}=s,{[L_{{\beta}/{\alpha}}(L_{1,\alpha})]}^{-{1}/{\beta}}=y$
has distribution $\mathrm{PD}(\alpha|s),$ not depending on $y.$ Using this,
the scaling property
$L_{{\beta}/{\alpha}}(s^{-\alpha})\overset{d}{=}L_{1,{\beta}/{\alpha}}\times
s^{-\beta},$ and elementary arguments to describe the joint density, it
follows that the expectation (2.6) can be expressed as

$\int_{0}^{\infty}\left[\int_{0}^{\infty}\mathbb{E}[\Omega(V_{\alpha})|T_{\alpha}=s]\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)f_{\alpha}(s)ds\right]h(y)f_{\beta}(y)dy,$

which can also be expressed as,

$\int_{0}^{\infty}\mathbb{E}[\Omega(V_{\alpha})|T_{\alpha}=s]\tilde{h}_{\frac{\beta}{\alpha}}(s)f_{\alpha}(s)ds$

for
$\tilde{h}_{\frac{\beta}{\alpha}}(s):=\mathbb{E}_{\frac{\beta}{\alpha}}\left[h\bigg{(}sT^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}}\bigg{)}\right]=\int_{0}^{\infty}\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)h(y)f_{\beta}(y)dy,$
yielding the results. ∎

Recall that for any $0<\beta<\sigma<\alpha<1,$
$\mathrm{FRAG}_{\alpha,-\beta}=\mathrm{FRAG}_{\alpha,-\sigma}\circ\mathrm{FRAG}_{\sigma,-\beta}.$

######  Corollary 2.1.

Suppose that $0<\beta<\sigma<\alpha<1,$ $V\sim\mathrm{PK}_{\beta}(h\cdot
f_{\beta})$ with $\mathbb{E}[h(T_{\beta})]=1.$ Then,
$V^{\prime}=\mathrm{FRAG}_{\sigma,-\beta}(V)\sim\mathrm{PK}_{\sigma}\bigg{(}\tilde{h}_{\frac{\beta}{\sigma}}\cdot
f_{\sigma}\bigg{)}$ where
$\tilde{h}_{\frac{\beta}{\sigma}}(s):=\mathbb{E}_{\frac{\beta}{\sigma}}\left[h\bigg{(}sT^{\frac{1}{\sigma}}_{\frac{\beta}{\sigma}}\bigg{)}\right].$
Hence,
$\tilde{V}=\mathrm{FRAG}_{\alpha,-\sigma}(V^{\prime})\sim\mathrm{PK}_{\alpha}\bigg{(}\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\bigg{)}$ where

$\tilde{h}_{\frac{\beta}{\alpha}}(v):=\mathbb{E}_{\frac{\sigma}{\alpha}}\left[\tilde{h}_{\frac{\beta}{\sigma}}\bigg{(}vT^{\frac{1}{\alpha}}_{\frac{\sigma}{\alpha}}\bigg{)}\right]=\mathbb{E}_{\frac{\beta}{\alpha}}\left[h\bigg{(}vT^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}}\bigg{)}\right].$

##  3 Duality via dependent coagulation

We now describe how to construct dependent coagulations to complete the dual
process of recovering $V\sim\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ from the
coagulation of
$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V)\sim\mathrm{PK}_{\alpha}(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}).$ Our results show how specification of $h(t)$ leads to a
prescription to identify the laws of $V,\tilde{V},Q$ without guess-work.

Recall that for the independent mass partitions
$(V_{\alpha},Q_{\frac{\beta}{\alpha}})$ described in (2.1), the process of
coagulation yields an inverse local time at $1$ for $V_{\beta}$ to be
$T_{\beta}(1)=T_{\alpha}(T_{{\beta}/{\alpha}}(1))\overset{d}{=}T_{\alpha}\times
T^{{1}/{\alpha}}_{{\beta}/{\alpha}}.$ For $\tilde{V}$ as described above, we
consider the dependent pair $(\tilde{V},Q)$ with joint law, say,
$\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h),$ characterized by

$\mathbb{E}[\Omega(\tilde{V},Q)]=\mathbb{E}^{(\frac{\beta}{\alpha},0)}_{(\alpha,0)}\left[\Omega\big{(}V_{\alpha},Q_{\frac{\beta}{\alpha}}\big{)}h\bigg{(}T_{\alpha}\big{(}T_{\frac{\beta}{\alpha}}(1)\big{)}\bigg{)}\right],$
(3.1)

with
$\mathbb{E}\big{[}h(T_{\beta}(1))\big{]}=\mathbb{E}\left[h\bigg{(}T_{\alpha}\big{(}T_{\frac{\beta}{\alpha}}(1)\big{)}\bigg{)}\right]=1$,
and the notation $\mathbb{E}^{(\frac{\beta}{\alpha},0)}_{(\alpha,0)}$
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 $(\tilde{V},Q)$ is given by

$\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h):=\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{PD}(\alpha|s)\mathrm{PD}({\beta}/{\alpha}|y)h\big{(}sy^{\frac{1}{\alpha}}\big{)}f_{\frac{\beta}{\alpha}}(y)f_{\alpha}(s)dyds.$
(3.2)

######  Remark 3.1.

For further clarity, we may use the notation
$\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h)=\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}\left(h\cdot(f_{\alpha},f_{\frac{\beta}{\alpha}})\right)$

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

$F_{\tilde{V}}(y)=\sum_{k=1}^{\infty}\tilde{V}_{k}\mathbb{I}_{\left\\{\tilde{U}_{k}\leq
y\right\\}}\qquad{\mbox{ and }}\qquad
G_{Q}(y)=\sum_{\ell=1}^{\infty}Q_{\ell}\mathbb{I}_{\left\\{U_{\ell}\leq
y\right\\}}.$ (3.3)

######  Remark 3.2.

It follows that when
$h(t)=t^{-\theta}/\mathbb{E}\big{[}T^{-\theta}_{\beta}\big{]}$ for
$\theta>-\beta,$ $\tilde{V}\sim\mathrm{PD}(\alpha,\theta)$ is independent of
$Q\sim\mathrm{PD}({\beta}/{\alpha},{\theta}/{\alpha}).$ Hence,
$F_{\tilde{V}}\overset{d}{=}F_{\alpha,\theta}$ and
$G_{Q}\overset{d}{=}G_{{\beta}/{\alpha},{\theta}/{\alpha}}$.

######  Proposition 3.1.

For $0<\beta<\alpha<1,$ let $(\tilde{V},Q)$ have a joint distribution,
$\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h),$ specified by (3.1) or
equivalently (3.2), such that
$\tilde{V}\sim\mathrm{PK}_{\alpha}\bigg{(}\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\bigg{)},$ and $(F_{\tilde{V}},G_{Q})$ are bridges defined in (3.3).
Let $V\in\mathcal{P}_{\infty}$ be the ranked masses of the bridge defined by
the composition $F_{V}:=F_{\tilde{V}}\circ G_{Q}.$ Then, $V$ is equivalent to
the coagulation of $\tilde{V}$ by $Q$ and there are the following properties.

  1. (i)

$V\sim\mathrm{PK}_{\beta}(h\cdot f_{\beta}).$

  2. (ii)

The marginal distribution of
$Q~{}\sim\mathrm{PK}_{\beta/\alpha}(\hat{h}_{\alpha}\cdot
f_{\frac{\beta}{\alpha}}),$ where

$\hat{h}_{\alpha}(y)=\mathbb{E}_{\alpha}[h(T_{\alpha}y^{\frac{1}{\alpha}})].$
(3.4)

and the corresponding inverse local time $\hat{T}_{1}$ has density
$\hat{h}_{{\alpha}}(y)f_{\frac{\beta}{\alpha}}(y)$.

  3. (iii)

The distribution of $\tilde{V}|\hat{T}_{1}=y$ is
$\mathrm{PK}_{\alpha}(h^{(y)}_{\alpha}\cdot f_{\alpha}),$ where

$h^{(y)}_{\alpha}(s)=\frac{h(sy^{\frac{1}{\alpha}})}{\mathbb{E}_{\alpha}[h(T_{\alpha}y^{\frac{1}{\alpha}})]}.$

###### Proof.

We first recall from (2.1) that under independent $\mathrm{PD}(\alpha,0)$ and
$\mathrm{PD}({\beta}/{\alpha},0)$ laws, the bridge $F_{\beta}:=F_{\alpha}\circ
G_{\frac{\beta}{\alpha}}$ 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 $(\tilde{V},Q)$ specified by (3.1), it follows that, for
$F_{V}:=F_{\tilde{V}}\circ G_{Q},$

$\mathbb{E}\big{[}\Omega(F_{\tilde{V}}\circ
G_{Q})\big{]}=\mathbb{E}\left[\Omega\big{(}F_{\alpha}\circ
G_{\frac{\beta}{\alpha}}\big{)}h\big{(}T_{\alpha}\big{(}T_{\frac{\beta}{\alpha}}(1)\big{)}\big{)}\right]$

showing that $F_{V}$ is a $\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ bridge and
thus $V\sim\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ in statement (i). Statements
(ii) and (iii) follow from straightforward usage of (3.2). ∎

The next result shows that $(\tilde{V},Q)|T_{V}=r$ is equivalent to
$(V_{\alpha},Q_{\frac{\beta}{\alpha}})|T_{\alpha}(T_{\frac{\beta}{\alpha}}(1))=r.$

######  Proposition 3.2.

For $0<\beta<\alpha<1,$ let $(\tilde{V},Q)$ have a joint distribution,
$\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h),$ specified by (3.1) or
equivalently (3.2), such that
$\tilde{V}\sim\mathrm{PK}_{\alpha}(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}),$ and $(F_{\tilde{V}},G_{Q})$ are bridges defined in (3.3). Let
$V\in\mathcal{P}_{\infty}$ be the ranked masses of the bridge defined by the
composition $F_{V}:=F_{\tilde{V}}\circ G_{Q}$, with inverse local time at $1$
denoted as $T_{V}$ with density $h(t)f_{\beta}(t).$ Then, by a change of
variable, the joint distribution
$\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h)$ can be expressed as

$\int_{0}^{\infty}\left[\int_{0}^{\infty}\mathrm{PD}(\alpha|ry^{-1/\alpha})\mathrm{PD}({\beta}/{\alpha}|y)\frac{f_{\alpha}(ry^{-\frac{1}{\alpha}})}{y^{\frac{1}{\alpha}}f_{\beta}(r)}f_{\frac{\beta}{\alpha}}(y)dy\right]h(r)f_{\beta}(r)dr.$
(3.5)

That is, the joint distribution of $(\tilde{V},Q)|T_{V}=r$ is equivalent to
$(V_{\alpha},Q_{\frac{\beta}{\alpha}})|T_{\alpha}(T_{\frac{\beta}{\alpha}}(1))=r$,
for all $h$, and it is given by

$\int_{0}^{\infty}\mathrm{PD}(\alpha|ry^{-1/\alpha})\mathrm{PD}\left(\left.{\beta}/{\alpha}\right|y\right)\frac{f_{\alpha}(ry^{-\frac{1}{\alpha}})}{y^{\frac{1}{\alpha}}f_{\beta}(r)}f_{\frac{\beta}{\alpha}}(y)dy.$

The next Corollary provides an answer to when or under which situation the
pair $(Q,V)$ specified by the coagulation $F_{{V}}=F_{\tilde{V}}\circ G_{Q}$
are in the same family of distributions, specifically, distributions of the
form $(\mathrm{PK}_{\frac{\beta}{\delta}}(\hat{h}_{\delta}\cdot
f_{\frac{\beta}{\delta}}),0<\beta<\delta<1),$ where
$\hat{h}_{\delta}(y)=\mathbb{E}_{\delta}[h(T_{\delta}y^{\frac{1}{\delta}})]$
as in (3.4).

######  Corollary 3.1.

For $0<\beta<\sigma<\alpha<1,$ consider the settings in Proposition 3.1 where
now
$(\tilde{V},Q)\sim\mathrm{P}^{\frac{\beta}{\sigma}}_{\frac{\sigma}{\alpha}}(\hat{h}_{\alpha}),$
and $V\in\mathcal{P}_{\infty}$ is obtained by the coagulation equivalent to
ranked masses of $F_{{V}}=F_{\tilde{V}}\circ G_{Q}.$ Then, the variables
$(V,\tilde{V},Q)$ have the following marginal (or conditional) distributions

  1. (i)

$V\sim\mathrm{PK}_{\frac{\beta}{\alpha}}\bigg{(}\hat{h}_{\alpha}\cdot
f_{\frac{\beta}{\alpha}}\bigg{)}$, for
$\hat{h}_{\alpha}(v)=\mathbb{E}_{\alpha}\bigg{[}h(T_{\alpha}v^{\frac{1}{\alpha}})\bigg{]}$.

  2. (ii)

$Q~{}\sim\mathrm{PK}_{\frac{\beta}{\sigma}}\bigg{(}\hat{h}_{\sigma}\cdot
f_{\frac{\beta}{\sigma}}\bigg{)},$ where $\hat{T}_{1}$ has density
$\hat{h}_{\sigma}\cdot f_{\frac{\beta}{\sigma}},$ for

$\hat{h}_{\sigma}(y)=\mathbb{E}_{\frac{\sigma}{\alpha}}\bigg{[}\hat{h}_{\alpha}(T_{\frac{\sigma}{\alpha}}y^{\frac{\alpha}{\sigma}})\bigg{]}=\mathbb{E}_{\sigma}[h(T_{\sigma}y^{\frac{1}{\sigma}})].$
(3.6)

  3. (iii)

The distribution of $\tilde{V}|\hat{T}_{1}=y$ is
$\mathrm{PK}_{\frac{\sigma}{\alpha}}\bigg{(}\big{(}\hat{h}_{\alpha}\big{)}^{(y)}_{\frac{\sigma}{\alpha}}\cdot
f_{\frac{\sigma}{\alpha}}\bigg{)},$ where

$\big{(}\hat{h}_{\alpha}\big{)}^{(y)}_{\frac{\sigma}{\alpha}}(s)=\mathbb{E}_{\alpha}\bigg{[}h\big{(}T_{\alpha}s^{\frac{1}{\alpha}}y^{\frac{1}{\sigma}}\big{)}\bigg{]}/\mathbb{E}_{\sigma}[h(T_{\sigma}y^{\frac{1}{\sigma}})].$

###### Proof.

The results follow from Proposition 3.1 and manipulating the distributional
properties of

$T_{\alpha}\times{\bigg{[}T_{\frac{\sigma}{\alpha}}\big{(}T_{\frac{\beta}{\sigma}}(1)\big{)}\bigg{]}}^{\frac{1}{\alpha}}\overset{d}{=}T_{\beta}.$

∎

######  Remark 3.3.

In Corollary 3.1,
$\tilde{V}=\mathrm{FRAG}_{\frac{\sigma}{\alpha},-\frac{\beta}{\alpha}}(V)$ has
distribution
$\mathrm{PK}_{\frac{\sigma}{\alpha}}\bigg{(}\widetilde{\big{(}\hat{h}_{\alpha}\big{)}}_{\frac{\sigma}{\alpha}}\cdot
f_{\frac{\sigma}{\alpha}}),$ where
$\widetilde{\big{(}\hat{h}_{\alpha}\big{)}}_{\frac{\sigma}{\alpha}}(s)=\mathbb{E}\bigg{[}h(T_{\alpha}s^{\frac{1}{\alpha}}T_{\frac{\beta}{\sigma}}^{\frac{1}{\sigma}})\bigg{]}$.

##  4 Gibbs partitions of $[n]$ derived from $\mathrm{FRAG}_{\alpha,-\beta}$

Recall from [42, 43] that when $V_{\beta}\sim\mathrm{PD}(\beta,0)$,
$V_{\beta}|L^{-{1}/{\beta}}_{1,\beta}=y$ is equivalent in distribution to
$V_{\beta}|T_{\beta}=y,$ and has the associated Gibbs partition of $[n]$
described by the $\mathrm{PD}(\beta|y)-\mathrm{EPPF},$

$p_{\beta}(n_{1},\ldots,n_{k}|y):=\frac{f^{(n-k\beta)}_{\beta,k\beta}(y)}{f_{\beta}(y)}p_{\beta}(n_{1},\ldots,n_{k}),$
(4.1)

where, as in[24, 25],

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

with, from [17, 42, 43],

$\mathbb{G}_{\beta}^{(n,k)}(t)=\frac{\beta^{k}t^{-n}}{\Gamma(n-k\beta)f_{\beta}(t)}\left[\int_{0}^{t}f_{\beta}(v)(t-v)^{n-k\beta-1}dv\right],$
(4.2)

and $f^{(n-k\beta)}_{\beta,k\beta}(y)$ being the conditional density of
$T_{\beta}|K^{[\beta]}_{n}=k$ corresponding to a random variable denoted as
$Y^{n-k\beta}_{\beta,k\beta},$ as otherwise described in (1.5) with $\beta$ in
place of $\alpha.$ Note, furthermore, as in [25], this means
$T_{\beta}:=T_{\alpha}(T_{\frac{\beta}{\alpha}}(1))\overset{d}{=}Y^{(n-K^{[\beta]}_{n}\beta)}_{\beta,K^{[\beta]}_{n}\beta},$
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.

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

Recall from Theorem 2.1 that the distribution of
$V_{\alpha}|L_{1,\beta}=y^{-\beta}$ is equivalent to that of
$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V)|L_{1,V}=y^{-\beta}$, with
distribution denoted
$\mathrm{PD}_{\alpha|\beta}(\alpha|y):=\mathrm{PK}_{\alpha}\bigg{(}\omega^{(y)}_{\frac{\beta}{\alpha},\beta}\cdot
f_{\alpha}\bigg{)}$ as in (2.4), where
$\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)$ 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 $\mathrm{PD}_{\alpha|\beta}(\alpha|y)$
Gibbs partition of $[n].$ We first note that since
$T_{\alpha}|K^{[\alpha]}_{n}=k$ is equivalent in distribution to
$Y^{n-k\alpha}_{\alpha,k\alpha}$ with density
$f^{(n-k\alpha)}_{\alpha,k\alpha},$ the $\mathrm{EPPF}$ can be expressed as

$\left[\int_{0}^{\infty}\omega^{(y)}_{\frac{\beta}{\alpha},\beta}(s)f^{n-k\alpha}_{\alpha,k\alpha}(s)ds\right]p_{\alpha}(n_{1},\ldots,n_{k}),$

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 [24] to express
$\omega^{(y)}_{\frac{\beta}{\alpha},\beta}$ in terms of Fox-$H$ functions
[37], leading to an expression for the EPPF in terms of Fox-$H$ functions in
the Appendix.

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

######  Theorem 4.1.

The $\mathrm{EPPF}$ of the $\mathrm{PD}_{\alpha|\beta}(\alpha|y)$ Gibbs
partition of $[n]$ can be expressed as

$p_{\alpha|\beta}(n_{1},\ldots,n_{k}|y):=\left[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{\beta}{\alpha},0}(j)\frac{f^{(n-j\beta)}_{\beta,j\beta}(y)}{f_{\beta}(y)}\right]p_{\alpha}(n_{1},\ldots,n_{k}),$
(4.3)

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

###### Proof.

The expression for the $\mathrm{EPPF}$ 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 (4.3) in terms of the marginal EPPF
$p_{\alpha}(n_{1},\ldots,n_{k})$ and the conditional density of
$T_{\beta}|K^{[\alpha]}_{n}=k.$ It remains to show that
$T_{\beta}|K^{[\alpha]}_{n}=k$ agrees with the expression in (4.3) as
indicated. Recall that $L_{1,\beta}=L_{\frac{\beta}{\alpha}}(L_{1,\alpha})$
and hence the corresponding inverse local time at $1$ is
$T_{\beta}:=T_{\beta}(1)=T_{\alpha}(T_{\frac{\beta}{\alpha}}(1))$
corresponding to the coagulation operation dictated by
$F_{\beta}=F_{\alpha}\circ G_{\frac{\beta}{\alpha}},$ as expressed in (2.1).
Sampling from $F_{\alpha}\circ G_{\frac{\beta}{\alpha}},$ that is, according
to variables
$\bigg{(}G^{-1}_{\frac{\beta}{\alpha}}(F^{-1}_{\alpha}(U^{\prime}_{i})),i\in[n]\bigg{)},$
it follows that this procedure produces a $\mathrm{PD}(\beta,0)$ partition of
$[n],$ with
$K^{[\beta]}_{n}\overset{d}{=}K^{[{\beta}/{\alpha}]}_{K^{[\alpha]}_{n}}$
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 $\mathrm{PD}(\frac{\beta}{\alpha},0)$ partition
of $[k],$ for $K^{[\alpha]}_{n}=k\leq n$. Now from [25],
$T_{\beta}\overset{d}{=}Y^{n-K^{[\beta]}_{n}\beta}_{\beta,K^{[\beta]}_{n}\beta}.$
Hence $T_{\beta}|K^{[\alpha]}_{n}=k$ is equivalent to
$Y^{(n-K^{[{\beta}/{\alpha}]}_{k}\beta)}_{\beta,K^{[{\beta}/{\alpha}]}_{k}\beta},$
which, using (4.1), leads to the description of the density of
$T_{\beta}|K^{[\alpha]}_{n}=k$ appearing in (4.3). ∎

######  Remark 4.1.

The result above is equivalent to showing that
$Y^{(n-K^{[{\beta}/{\alpha}]}_{k}\beta)}_{\beta,K^{[{\beta}/{\alpha}]}_{k}\beta}\overset{d}{=}Y^{(n-k\alpha)}_{\alpha,k\alpha)}\times
T^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}},$ which can be deduced directly
using the subordinator representation [25, Theorem 2.1 and Proposition 2.1]
and decompositions of beta variables.

We now describe the distribution of the number of blocks and its limiting
behavior.

######  Corollary 4.1.

Consider the $\mathrm{EPPF}$ of a $\mathrm{PD}_{\alpha|\beta}(\alpha|y)$
partition of $[n]$ as in (4.3), for each $y>0.$ Let $\hat{K}_{n}(y)$ denote
the corresponding random number of unique blocks. Then, for $k=1,\ldots,n,$

$\mathbb{P}\bigg{(}K^{[\alpha]}_{n}=k\left|L_{1,\beta}=y^{-\beta}\right.\bigg{)}=\mathbb{P}(\hat{K}_{n}(y)=k)=\left[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{\beta}{\alpha},0}(j)\frac{f^{(n-j\beta)}_{\beta,j\beta}(y)}{f_{\beta}(y)}\right]\mathbb{P}^{(n)}_{\alpha,0}(k),$

and, as $n\rightarrow\infty,$
$n^{-\alpha}\hat{K}_{n}(y)\overset{a.s.}{\rightarrow}\hat{Z}_{\alpha|\beta}(y),$
where $\hat{Z}_{\alpha|\beta}(y)$ is equivalent in distribution to that of
$L_{1,\alpha}|L^{-{1}/{\beta}}_{1,\beta}=y,$ for
$L_{1,\beta}=L_{\frac{\beta}{\alpha}}(L_{1,\alpha}).$

###### Proof.

The distribution follows as a special case of known properties of the
distribution of the number of blocks of Gibbs partitions, and is otherwise
easy to verify directly from the EPPF. The limiting distribution follows as a
special case of [42, Proposition 13]. ∎

###  4.2 EPPF of
$\mathrm{FRAG}_{\alpha,-\beta}(V)\sim\mathrm{PK}_{\alpha}(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha})$

Recall from [17, 42], see also [25], that if $V\sim\mathrm{PK}_{\beta}(h\cdot
f_{\beta})$ with $\mathbb{E}[h(T_{\beta})]=1$, then the $\mathrm{EPPF}$ of its
associated Gibbs partition of $[n]$ is described as

$p^{[\nu]}_{\beta}(n_{1},\ldots,n_{k})={\Psi}^{[\beta]}_{n,k}\times
p_{\beta}(n_{1},\ldots,n_{k}),$ (4.4)

where
$\Psi^{[\beta]}_{n,k}=\mathbb{E}_{\beta}[h(T_{\beta})|K^{[\beta]}_{n}=k]$ and,
for clarity, $K^{[\beta]}_{n}$ is the number of blocks of a
$\mathrm{PD}(\beta,0)$ partition of $[n].$

Theorem 4.1 leads to the EPPF corresponding to
$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V)\sim\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\right),$ or any variable in $\mathcal{P}_{\infty}$ having the same
distribution.

######  Proposition 4.1.

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

$\left[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{\beta}{\alpha},0}(j)\Psi^{[\beta]}_{n,j}\right]p_{\alpha}(n_{1},\ldots,n_{k}),$
(4.5)

and there is the identity, for
$T_{\beta}:=T_{\beta}(1)=T_{\alpha}(T_{\frac{\beta}{\alpha}}(1)),$

$\mathbb{E}_{\alpha}\left[\tilde{h}_{\frac{\beta}{\alpha}}(T_{\alpha})\left|K^{[\alpha]}_{n}=k\right.\right]=\mathbb{E}\left[h(T_{\beta})\left|K^{[\alpha]}_{n}=k\right.\right]=\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{\beta}{\alpha},0}(j)\Psi^{[\beta]}_{n,j}.$

###### Proof.

The $\mathrm{EPPF}$ is equivalent to
$\int_{0}^{\infty}p_{\alpha|\beta}(n_{1},\ldots,n_{k}|y)h(y)f_{\beta}(y)dy$,
and hence the result follows from (4.3) in Theorem 4.1. ∎

######  Remark 4.2.

The expression in (4.5) provides a description of any mass partition with
distribution $\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\right)$ where
$\tilde{h}_{\frac{\beta}{\alpha}}(v):=\mathbb{E}_{\frac{\beta}{\alpha}}\left[h\big{(}vT^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}}\big{)}\right],$
regardless of whether or not it actually arises from a fragmentation
operation.

As a check, in the case where $(P_{\ell,0})\sim\mathrm{PD}(\beta,\theta),$
(4.5) must satisfy

$\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{\beta}{\alpha},0}(j)\frac{\Gamma\big{(}\frac{\theta}{\beta}+j\big{)}}{\Gamma\big{(}\frac{\theta}{\beta}+1\big{)}\Gamma(j)}=\frac{\Gamma\big{(}\frac{\theta}{\alpha}+k\big{)}}{\Gamma\big{(}\frac{\theta}{\alpha}+1\big{)}\Gamma(k)}.$
(4.6)

However, (4.6) is verified since it agrees with [43, exercise 3.2.9, p.66],
with $k$ in place of $n.$ There is the following Corollary in the case of
$\beta/\alpha=\frac{1}{2}$.

######  Corollary 4.2.

Specializing Theorem 4.1 to the case of $\beta/\alpha=\frac{1}{2},$ where
$V\sim\mathrm{PK}_{\frac{\alpha}{2}}\left(h\cdot f_{\frac{\alpha}{2}}\right),$
and
$\Psi^{[{\alpha}/{2}]}_{n,j}=\mathbb{E}_{\alpha/2}\left[h(T_{\frac{\alpha}{2}})\left|K^{[{\alpha}/{2}]}_{n}=j\right.\right],$
the $\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{1}{2}}\cdot f_{\alpha}\right)$
$\mathrm{EPPF}$ in (4.5) becomes

$\left[\sum_{j=1}^{k}{{2k-j-1}\choose{k-1}}2^{j+1-2k}\Psi^{[{\alpha}/{2}]}_{n,j}\right]p_{\alpha}(n_{1},\ldots,n_{k}).$
(4.7)

###  4.3 Generating $\mathrm{PD}_{\alpha|\beta}(\alpha|y),$ partitions via
$\mathrm{PD}(\alpha,-\beta)$ fragmentation of partitions

While the EPPF’s (4.3) and (4.5) are quite interesting from various
perspectives, it is not entirely necessary to employ them directly to obtain
random partitions from $\mathrm{PD}_{\alpha|\beta}(\alpha|y)$ and
$\mathrm{PK}_{\alpha}(\tilde{h}_{\frac{\beta}{\alpha}}\cdot f_{\alpha}).$ A
two-stage sampling scheme may be employed utilizing the dual partition-based
interpretation of the $\mathrm{Frag}_{\alpha,-\beta}$ operator. The following
scheme can be deduced from Bertoin [6], see also [41, 43].

  * 1.

Generate $n$ iid $\mathrm{PD}(\alpha,-\beta)$ partitions of $[n],$ say,
$\mathcal{A}_{1},\ldots,\mathcal{A}_{n},$ where, for each $i,$
$\mathcal{A}_{i}:=\\{A^{(i)}_{1},\ldots,A^{(i)}_{M^{(i)}_{n}}\\}$ with
$M^{(i)}_{n}$ blocks.

  * 2.

Independent of this, for each fixed $y,$ generate a $\mathrm{PD}(\beta|y)$
partition of $[n]$, say, $\\{C_{1},\ldots,C_{K_{n}(y)}\\}$, where $K_{n}(y)$
denotes the number of blocks.

  * 3.

For $i=1,\ldots,K_{n}(y)$, Consider the pairs $(C_{i},\mathcal{A}_{i})$ and
fragment $C_{i}$ by $\mathcal{A}_{i}$ according to

$\mathcal{C}_{i}=\left\\{C_{i,j}:=C_{i}\cap A^{(i)}_{j}:C_{i}\cap
A^{(i)}_{j}\neq\emptyset,j\in\big{\\{}1,\ldots,M^{(i)}_{n}\big{\\}}\right\\}.$

  * 4.

The collection $\\{C_{i,j}\in\mathcal{C}_{i}:i\in[K_{n}(y)]\\}$ (arranged
according to the least element) constitutes a
$\mathrm{PD}_{\alpha|\beta}(\alpha|y)$ partition of $[n],$ with

$\hat{K}_{n}(y)\overset{d}{=}\sum_{i=1}^{K_{n}(y)}K^{(i)}_{|C_{i}|},$ (4.8)

where $K^{(i)}_{|C_{i}|}\overset{d}{=}|\mathcal{C}_{i}|,$ and given $C_{i},$
is equivalent to the number of blocks in a $\mathrm{PD}(\alpha,-\beta)$
partition of $C_{i},$ conditionally independent for
$i=1,\ldots,\hat{K}_{n}(y).$

  * 5.

Replace Step 2 with a $\mathrm{PK}_{\beta}(h\cdot f_{\beta})$ partition of
$[n]$ to obtain a
$\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\right)$ partition of $[n].$

######  Remark 4.3.

The scheme above requires sampling of a $\mathrm{PD}(\beta|y)$ partition of
$[n].$ The relevant results of [24] show that this is the easiest when $\beta$
is a rational number. In that case, $\mathbb{G}^{(n,k)}_{\beta}(y)$ has a
tractable representation in terms of Meijer $G$ functions. So, this applies
to, in particular, $(P_{k,1})\sim\mathrm{PD}_{\alpha|\frac{m}{r}}(\alpha|y)$
for every $\alpha>{m}r,$ where $m<r$ are co-prime positive integers. We look
at perhaps the most remarkable case,
$\mathrm{PD}_{\alpha|\frac{1}{2}}(\alpha|y),$ in the forthcoming section 4.5.

###  4.4 Representations of the
$\mathrm{PK}_{\alpha}(\tilde{h}_{\frac{\beta}{\alpha}}\cdot f_{\alpha})$
$\alpha$-diversity, $D$ and $T$ interval partitions, and fixed point equations

The sampling scheme above shows that the number of blocks of a
$\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\right)$, say, $\hat{K}_{n},$ of partition of $[n]$ satisfies

$\hat{K}_{n}\overset{d}{=}\sum_{i=1}^{K_{n}}K^{(i)}_{|C_{i}|},$ (4.9)

where $K_{n}$ is the number of blocks in a $\mathrm{PK}_{\beta}(h\cdot
f_{\beta})$ partition of $[n]$. Hence, by standard results for exchangeable
partitions, see for instance [43], as $n\rightarrow\infty$,
$(|{C_{i}|/n,i\in[K_{n}]})\overset{d}{\rightarrow}((\tilde{P}_{k,\beta}))$ for
$((\tilde{P}_{k,\beta}))$ the size-biased re-arrangement of
$P_{\beta}=((P_{k,\beta}))\sim\mathrm{PK}_{\beta}({h}\cdot f_{\beta}),$ and
from [42],
$|C_{i}|^{-\alpha}K^{(i)}_{|C_{i}|}\overset{a.s.}{\rightarrow}Z^{(i)}_{\alpha,-\beta}\sim\mathrm{ML}(\alpha,-\beta).$
Expressed in other terms, as $n\rightarrow\infty,$

$\left(\frac{|C_{j}|}{n},\frac{K^{(j)}_{|C_{j}|}}{n^{\alpha}}\right)\overset{d}{\rightarrow}\left(\tilde{P}_{j,\beta},\tilde{P}^{\alpha}_{j,\beta}Z^{(j)}_{\alpha,-\beta}\right)$
(4.10)

as $j$ varies. The result (4.10), and its ranked version involving
$P_{\beta}=((P_{j,\beta}))\sim\mathrm{PK}_{\beta}({h}\cdot f_{\beta}),$ can be
interpreted as extensions of descriptions in [3, Theorems 6,7, Propositions
10,11], see also [43, Chapter 9], to the present setting with general $\beta,$
in place of $\beta=0,$ for the sequence of paired lengths and local times of
$D-$ and $T-\mathrm{partition}$ intervals. Hence the next results may be
thought of in those terms.

######  Proposition 4.2.

Let $\hat{K}_{n},$ with probability mass function, for $k=1,2,\ldots n,$

$\mathbb{P}(\hat{K}_{n}=k)=\mathbb{E}_{\alpha}\left[\left.\tilde{h}_{\frac{\beta}{\alpha}}(T_{\alpha})\right|K^{[\alpha]}_{n}=k\right]\mathbb{P}_{\alpha}^{(n)}(k)=\left[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{\beta}{\alpha},0}(j)\Psi^{[\beta]}_{n,j}\right]\mathbb{P}_{\alpha}^{(n)}(k),$

denote the number of blocks in a
$\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\right)$ partition of $[n],$ as otherwise specified in (4.5) of
Proposition 4.1. Then,
$n^{-\alpha}\hat{K}_{n}\overset{a.s.}{\rightarrow}\hat{Z},$ where $\hat{Z}$ is
the $\alpha$-diversity and has density
$\mathbb{E}_{\frac{\beta}{\alpha}}\left[h(s^{-\frac{1}{\alpha}}T^{\frac{1}{\alpha}}_{\frac{\beta}{\alpha}})\right]g_{\alpha}(s).$
Furthermore, there is the distributional identity

$\hat{Z}\overset{d}{=}\sum_{k=1}^{\infty}P^{\alpha}_{k,\beta}Z^{(k)}_{\alpha,-\beta}\overset{d}{=}\sum_{k=1}^{\infty}\tilde{P}^{\alpha}_{k,\beta}Z^{(k)}_{\alpha,-\beta},$

where $P_{\beta}=((P_{k,\beta}))\sim\mathrm{PK}_{\beta}({h}\cdot f_{\beta}),$
and $(\tilde{P}_{k,\beta})$ is its size-biased re-arrangement, independent of
$((Z^{(k)}_{\alpha,-\beta}))\overset{iid}{\sim}\mathrm{ML}(\alpha,-\beta).$

Recall the (pointwise) identity from size-biased sampling
$V_{\alpha,\theta}\sim\mathrm{PD}(\alpha,\theta),$ for $\theta>-\alpha,$ with,
for each $\ell=1,2,\ldots,$

$Z_{\alpha,\theta}=Z_{\alpha,\theta+\ell\alpha}\prod_{i=1}^{\ell}W^{\alpha}_{i}=Z_{\alpha,\theta+\ell}\prod_{i=1}^{\ell}B_{i},$
(4.11)

where
$Z_{\alpha,\theta+\ell\alpha}:=T^{-\alpha}_{\alpha,\theta+\ell\alpha}\sim\mathrm{ML}(\alpha,\theta+\ell\alpha)$
is independent of the independent collection $(W_{1},\ldots,W_{\ell}),$ with
each $W_{j}\sim\mathrm{Beta}(\theta+j\alpha,1-\alpha)$, and
$Z_{\alpha,\theta+\ell}\sim\mathrm{ML}(\alpha,\theta+\ell)$ is independent of
the independent collection $(B_{1},\ldots,B_{\ell}),$ with each
$B_{j}\sim\mathrm{Beta}(\frac{\theta+j\alpha}{\alpha},\frac{1-\alpha}{\alpha}).$
See [14, 25, 46].

Applying the $\mathrm{FRAG}_{\alpha,-\beta}$ operator, in the case of the
$\mathrm{PD}$ setting established in [41], to
$V_{\beta,\ell\alpha-\beta}\sim\mathrm{PD}(\beta,{\ell\alpha-\beta}),$ and
$V_{\beta,\ell-\beta}\sim\mathrm{PD}(\beta,{\ell-\beta}),$ for
$\ell=1,2,\ldots,$ yields mass partitions
$V_{\alpha,\ell\alpha-\beta}\sim\mathrm{PD}(\alpha,{\ell\alpha-\beta}),$ and
$V_{\alpha,\ell-\beta}\sim\mathrm{PD}(\alpha,{\ell-\beta}),$ Proposition 4.2
and special cases of (4.11), with $\theta=-\beta,$ leads to the following
fixed point equations in the sense of [1, 18, 27], for certain generalized
Mittag-Leffler variables, which we believe are new,

######  Proposition 4.3.

For $0\leq\beta<\alpha<1,$ consider the identity in (4.11) for the case where
$\theta=-\beta,$ let $\left(\prod_{i=1}^{\ell}W_{i,k}^{\alpha},k\geq 1\right)$
and $\left(\prod_{i=1}^{\ell}B_{i,k},k\geq 1\right)$ denote iid collections of
variables having distribution equivalent to $\prod_{i=1}^{\ell}W^{\alpha}_{i}$
and $\prod_{i=1}^{\ell}B_{i},$ respectively, and, independent of other
variables, let $\left(Z^{(k)}_{\alpha,\ell\alpha-\beta},k\geq 1\right)$ and
$\left(Z^{(k)}_{\alpha,\ell-\beta},k\geq 1\right)$ denote iid collections of
variables with each component having distribution
$\mathrm{ML}(\alpha,\ell\alpha-\beta)$ and $\mathrm{ML}(\alpha,\ell-\beta)$,
respectively. Then, for each $\ell=1,2,\ldots$, there are the following fixed
point equations,

  1. (i)

for $(\tilde{P}_{k,\beta})\sim\mathrm{GEM}(\beta,{\ell\alpha-\beta}),$ and
$Z_{\alpha,\ell\alpha-\beta}\sim\mathrm{ML}(\alpha,\ell\alpha-\beta),$
equivalent in distribution to the $\alpha$-diversity of
$(P_{k,\alpha})\sim\mathrm{PD}(\alpha,\alpha\ell-\beta),$

$Z_{\alpha,\ell\alpha-\beta}\overset{d}{=}\sum_{k=1}^{\infty}\tilde{P}^{\alpha}_{k,\beta}\prod_{i=1}^{\ell}W^{\alpha}_{i,k}Z^{(k)}_{\alpha,\ell\alpha-\beta},$

and, hence,
$\mathbb{E}\left[\sum_{k=1}^{\infty}\tilde{P}^{\alpha}_{k,\beta}\prod_{i=1}^{\ell}W^{\alpha}_{i,k}\right]=1.$

  2. (ii)

For $(\tilde{P}_{k,\beta})\sim\mathrm{GEM}(\beta,{\ell-\beta}),$ and
$Z_{\alpha,\ell-\beta}\sim\mathrm{ML}(\alpha,\ell-\beta),$ equivalent in
distribution to the $\alpha$-diversity of
$(P_{k,\alpha})\sim\mathrm{PD}(\alpha,\ell-\beta),$

$Z_{\alpha,\ell-\beta}\overset{d}{=}\sum_{k=1}^{\infty}\tilde{P}^{\alpha}_{k,\beta}\prod_{i=1}^{\ell}B_{i,k}Z^{(k)}_{\alpha,\ell-\beta},$

and, hence,
$\mathbb{E}\left[\sum_{k=1}^{\infty}\tilde{P}^{\alpha}_{k,\beta}\prod_{i=1}^{\ell}B_{i,k}\right]=1.$

For clarity, for $(\tilde{P}_{k,\beta})$ in Proposition 4.3, one has
$(\tilde{P}_{k,\beta})\overset{d}{=}(R_{k}\prod_{j=1}^{k-1}(1-R_{j}),k\geq
1),$ where in the case of
$(\tilde{P}_{k,\beta})\sim\mathrm{GEM}(\beta,{\ell\alpha-\beta}),$ the
$((R_{k}\overset{ind}{\sim}\mathrm{Beta}(1-\beta,\ell\alpha+(k-1)\beta))$, and
for $(\tilde{P}_{k,\beta})\sim\mathrm{GEM}(\beta,{\ell-\beta}),$ the
$((R_{k}\overset{ind}{\sim}\mathrm{Beta}(1-\beta,\ell+(k-1)\beta))$, for all
$0\leq\beta<1$. When $\beta=0,$ one has special cases corresponding to the
fragmentation results
$V_{\alpha,\ell\alpha}=\mathrm{FRAG}_{\alpha,0}(V_{0,\ell\alpha}),$ and
$V_{\alpha,\ell}=\mathrm{FRAG}_{\alpha,0}(V_{0,\ell}),$ see [43, Chapter 5, p.
119].

###  4.5 $\mathrm{PD}(\alpha,-\frac{1}{2})$ Fragmentation of a Brownian
excursion partition conditioned on its local time

Following Pitman [42, Section 8] and [43, Section 4.5, p.90], let
$(P_{\ell,0})\sim\mathrm{PD}\big{(}\frac{1}{2},0\big{)}$ denote the ranked
excursion lengths of a standard Brownian motion $B:=(B_{t}:t\in[0,1])$, with
corresponding local time at $0$ up till time $1$ given by
$L_{1}\overset{d}{=}\big{(}2T_{\frac{1}{2}}\big{)}^{-\frac{1}{2}}\overset{d}{=}|B_{1}|.$
Then, it follows that $(P_{\ell,0})|L_{1}=\lambda$ has a
$\mathrm{PD}(\frac{1}{2}|\frac{1}{2}\lambda^{-2})$ distribution. Furthermore,
with respect to $(P_{\ell}(s))\sim\mathrm{PD}(\frac{1}{2}|\frac{1}{2}s^{-2}),$
we describe the special $\beta=\frac{1}{2}$ explicit case of the Gibbs
partitions (EPPF) of $[n]$ in terms of Hermite functions as derived in [42],
see also [43, Section 4.5], as

$p_{\frac{1}{2}}\left(n_{1},\ldots,n_{k}\left|\frac{1}{2}s^{-2}\right.\right)=s^{k-1}\tilde{H}_{k+1-2n}(s)\frac{\Gamma(n)}{2^{1-n}\Gamma(k)}p_{\frac{1}{2}}(n_{1},\ldots,n_{k}),$
(4.12)

where, for $U(a,b,c)$ a confluent hypergeometric function of the secondkind
(see [35, p.263]),

$\tilde{H}_{-2q}(s)=2^{-q}U\left(q,\frac{1}{2},\frac{s^{2}}{2}\right)=\sum_{\ell=0}^{\infty}\frac{{(-s)}^{\ell}}{\ell!}\frac{\Gamma(q+\frac{\ell}{2})}{2\Gamma(2q)}2^{q+\frac{\ell}{2}}$

is a Hermite function of index $-2q.$ That is, to say

$\mathbb{G}^{(n,k)}_{\frac{1}{2}}\bigg{(}\frac{1}{2}s^{-2}\bigg{)}=2^{n-k}s^{k-1}\tilde{H}_{k+1-2n}(s).$
(4.13)

######  Proposition 4.4.

Suppose that
$P_{1/2}(s):=(P_{\ell}(s))\sim\mathrm{PD}(\frac{1}{2}|\frac{1}{2}s^{-2}).$
Then, for $\alpha>1/2,$

$P_{\alpha}(s)=\mathrm{FRAG}_{\alpha,-\frac{1}{2}}(P_{1/2}(s))\sim\mathrm{PD}_{\alpha|\frac{1}{2}}\left(\alpha\left|\frac{1}{2}s^{-2}\right.\right),$

with corresponding $\mathrm{EPPF}$ expressed in terms of a mixture of Hermite
functions,

$\left[\sum_{j=1}^{k}\mathbb{P}^{(k)}_{\frac{1}{2\alpha},0}(j)2^{n-1}s^{j-1}\tilde{H}_{j+1-2n}(s)\frac{\Gamma(n)}{\Gamma(j)}\right]p_{\alpha}(n_{1},\ldots,n_{k}).$
(4.14)

###### Proof.

The result follows as a special case of Theorem 2.1 and Theorem 4.1, and
otherwise using the explicit form of the EPPF in (4.12). ∎

######  Remark 4.4.

In order to obtain a partition of $[n]$ corresponding to the EPPF in (4.14),
one can sample from (4.12) via the prediction rules indicated in [42, eqs.
(111) and (112)], or otherwise employ the scheme described in Section 4.3.

Now recall from [42, Proposition 14], equivalently [2, Corollary 5], that
$(\tilde{P}_{\ell}(s)),$ the size biased re-arrangement of
$(P_{\ell}(s))\sim\mathrm{PD}(\frac{1}{2}|\frac{1}{2}s^{-2}),$ has a version
with the explicit representation for each $j\geq 1,$ jointly and pointwise,

$\tilde{P}_{j}(s)=\frac{s^{2}}{s^{2}+S_{j-1}}-\frac{s^{2}}{s^{2}+S_{j}}$
(4.15)

for $S_{j}:=\sum_{i=1}^{j}\tilde{X}_{i}$, where $\tilde{X}_{i}$ are
independent and identically distributed variables like
$B^{2}_{1}\sim\chi^{2}_{1}$ for $B_{1}\sim\mathrm{N}(0,1)$ a standard Gaussian
variable. Now, in view of the results and discussions in Sections 4.3 and 4.4,
we easily obtain the following interesting result.

######  Proposition 4.5.

Let
$\hat{K}_{n}(s^{-2}/2)\overset{d}{=}\sum_{i=1}^{K_{n}(s^{-2}/2)}K^{(i)}_{|C_{i}|}$
denote the number of blocks in a
$\mathrm{PD}_{\alpha|\frac{1}{2}}(\alpha|\frac{1}{2}s^{-2})$ partition of
$[n],$ as can otherwise be generated according to the scheme in Section 4.3,
by using a $\mathrm{PD}(\alpha,-1/2)$ fragmentation of a
$\mathrm{PD}(\frac{1}{2}|\frac{1}{2}s^{-2})$ partition of $[n]$ with blocks
$\\{C_{i},i\in[K_{n}(s^{-2}/2)]\\}.$ $\hat{K}_{n}(s^{-2}/2)$ is as otherwise
described in Corollary 4.1 with corresponding $\alpha$-diversity
$\hat{Z}_{\alpha|1/2}(s^{-2}/2).$ Then, for $(\tilde{P}_{j}(s),j\geq 1)$ as in
(4.15),

$\hat{Z}_{\alpha|1/2}\bigg{(}\frac{1}{2}s^{-2}\bigg{)}\overset{d}{=}\sum_{j=1}^{\infty}[\tilde{P}_{j}(s)]^{\alpha}Z^{(j)}_{\alpha,-1/2}.$

Furthermore, as $n\rightarrow\infty,$

$\left(\frac{|C_{j}|}{n},\frac{K^{(j)}_{|C_{j}|}}{n^{\alpha}}\right)\overset{d}{\rightarrow}\left(\tilde{P}_{j}(s),[\tilde{P}_{j}(s)]^{\alpha}Z^{(j)}_{\alpha,-1/2}\right)$

jointly for $j=1,2,\ldots,$ where $\hat{Z}_{\alpha|1/2}(\frac{1}{2}s^{-2})$ is
equivalent in distribution to that of $L_{1,\alpha}|L_{1,1/2}=\sqrt{2}s$, and
$L_{1,1/2}=L_{\frac{1}{2\alpha}}(L_{1,\alpha}).$

###  4.6 $\mathrm{FRAG}_{\alpha,-\beta}$ 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 [25], see also [28]. Recall that for
$\lambda>0,$ the Laplace transform of
$L_{1,\beta}\overset{d}{=}T^{-\beta}_{\beta}$ equates with the Mittag-Leffler
function, see for instance [20], expressed as,

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

Let $(N(s),s\geq 0)$ denote a standard Poisson process, where
$\mathbb{E}[N(s)]=s,$ and consider the mixed Poisson process
$(N(tL_{1,\beta}),t\geq 0).$ Then, as shown in [25], for
$V_{\beta}\sim\mathrm{PD}(\beta,0),$ $V_{\beta}|N(\lambda L_{1,\beta})=0$
corresponds in distribution to

$V_{\beta}(\lambda)\sim\int_{0}^{\infty}\mathrm{PD}(\beta|t)\frac{{\mbox{e}}^{-\lambda
t^{-\beta}}}{\mathrm{E}_{\beta,1}(-\lambda)}f_{\beta}(t)dt,$ (4.16)

that is, to say a stable Poisson-Kingman distribution with index $\beta$ and
$h(t)={\mbox{e}}^{-\lambda t^{-\beta}}/\mathrm{E}_{\beta,1}(-\lambda).$

######  Proposition 4.6.

Let $V_{\beta}(\lambda):=(V_{k,\beta}(\lambda))$ have distribution specified
in (4.16), and otherwise consider the setting in Theorem 2.1.

  1. (i)

$\tilde{V}_{\alpha}(\lambda)=\mathrm{FRAG}_{\alpha,-\beta}(V_{\beta}(\lambda))\sim\mathrm{PK}_{\alpha}\left(\tilde{h}_{\frac{\beta}{\alpha}}\cdot
f_{\alpha}\right)$ where

$\tilde{h}_{\frac{\beta}{\alpha}}(v)=\frac{\mathrm{E}_{\frac{\beta}{\alpha},1}(-\lambda
v^{-\beta})}{\mathrm{E}_{\beta,1}(-\lambda)}.$

  2. (ii)

Let $L_{1,\alpha}(\lambda)\overset{d}{=}T^{-\alpha}_{\alpha}(\lambda)$ denote
the corresponding local time/$\alpha$-diversity with density
${\mathrm{E}_{\frac{\beta}{\alpha},1}(-\lambda
s^{\frac{\beta}{\alpha}})}g_{\alpha}(s)/{\mathrm{E}_{\beta,1}(-\lambda)},$
then

$L_{1,\alpha}(\lambda)\overset{d}{=}\sum_{j=1}^{\infty}[V_{j,\beta}(\lambda)]^{\alpha}Z^{(j)}_{\alpha,-\beta},$

where
$((Z^{(j)}_{\alpha,-\beta}))\overset{iid}{\sim}\mathrm{ML}(\alpha,-\beta),$
independent of $V_{\beta}(\lambda)$.

##  5 Duals in the generalized Mittag-Leffler class

In Section 4.6, we examined the fragmentation of the Mittag-Leffler variable,
$V_{\beta}(\lambda)$ having distribution equivalent to $V_{\beta}|N(\lambda
L_{1,\beta})=0.$ The extension to more generalized Mittag-Leffler classes
could proceed as in [25], by directly conditioning on $V_{\beta}$ as
$V_{\beta}|N(\lambda L_{1,\beta})=j,$ for fixed $j=0,1,2,\ldots.$ Here we show
that the corresponding $\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(h)$ joint
distribution can arise directly from
$(V_{\alpha},Q_{\frac{\beta}{\alpha}})|N(\lambda L_{1,\beta})=j,$ which, has
not been previously studied. In order to describe this, we first recall the
Laplace transform of the $(P_{\ell})\sim\mathrm{PD}(\beta,\theta)$ local time
at $1$ variable, or $\beta$-diversity, say
$\hat{L}_{\beta,\theta}\overset{d}{=}T^{-\beta}_{\beta,\theta}\sim\mathrm{ML}(\beta,\theta),$
with density
$g_{\beta,\theta}(s)=s^{\theta/\beta}g_{\beta}(s)/\mathbb{E}[T^{-\theta}_{\beta}],$
as given in [28, Section 3], and [25, Section 4.1],

$\mathbb{E}\big{[}{\mbox{e}}^{-\lambda
T^{-\beta}_{\beta,\theta}}\big{]}=\mathbb{E}\big{[}{\mbox{e}}^{-\lambda^{1/\beta}X_{\beta,\theta}}\big{]}=\mathrm{E}^{(\frac{\theta}{\beta}+1)}_{\beta,\theta+1}(-\lambda),$
(5.1)

where $X_{\beta,\theta}:=T_{\beta}/T_{\beta,\theta}$ is the Lamperti variable
studied in [28], and

$\mathrm{E}^{(\frac{\theta}{\beta}+1)}_{\beta,\theta+1}(-\lambda)=\sum_{\ell=0}^{\infty}\frac{{(-\lambda)}^{\ell}}{\ell!}\frac{\Gamma(\frac{\theta}{\beta}+1+\ell)\Gamma(\theta+1)}{\Gamma(\frac{\theta}{\beta}+1)\Gamma(\beta\ell+\theta+1)},\qquad\theta>-\beta,$
(5.2)

and from [25, Proposition 4.4] there is the density

$g^{(0)}_{\beta,\theta+m\beta}(s|\lambda)=\mathbb{P}(\hat{L}_{\beta,\theta}\in
ds|N(\lambda\hat{L}_{\beta,\theta})=j)/ds=\frac{{\mbox{e}}^{-\lambda
s}g_{\beta,\theta+j\beta}(s)}{\mathrm{E}^{(\frac{\theta}{\beta}+j+1)}_{\beta,\theta+j\beta+1}(-\lambda)}.$

It follows that for $(P_{\ell})\sim\mathrm{PD}(\beta,\theta),$
$(P_{\ell})|N(\lambda\hat{L}_{\beta,\theta})=j\sim\mathbb{L}^{(0)}_{\beta,\theta+j\beta}(\lambda),$
such that,

$\mathbb{L}^{(0)}_{\beta,\theta+j\beta}(\lambda):=\int_{0}^{\infty}\mathrm{PD}(\beta|s^{-\frac{1}{\beta}})\,g^{(0)}_{\beta,\theta+j\beta}(s|\lambda)\,ds,$
(5.3)

and, hence, in this case,

$h(t):=\vartheta^{(\lambda)}_{\beta,\theta+j\beta}(t)=\frac{{\mbox{e}}^{-\lambda
t^{-\beta}}t^{-\theta-j\beta}}{\mathrm{E}^{(\frac{\theta}{\beta}+j+1)}_{\beta,\theta+j\beta+1}(-\lambda)\mathbb{E}[T_{\beta}^{-\theta-j\beta}]}.$

Setting $\theta=0,$ it follows for the variables
$(V_{\alpha},Q_{\frac{\beta}{\alpha}}),$ and $V_{\beta}$ defined according to
coag/frag duality in [8, 41], that the distribution of
$(V_{\alpha},Q_{\frac{\beta}{\alpha}})|N(\lambda
L_{1,\beta})=j\sim\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(\vartheta^{(\lambda)}_{\beta,j\beta}),$
which can be expressed

$\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{PD}(\alpha|s)\mathrm{PD}({\beta}/{\alpha}|y)\frac{{\mbox{e}}^{-\lambda
s^{-\beta}y^{\frac{-\beta}{\alpha}}}s^{-j\beta}y^{-\frac{j\beta}{\alpha}}}{\mathrm{E}^{(j+1)}_{\beta,j\beta+1}(-\lambda)\mathbb{E}[T_{\beta}^{-j\beta}]}f_{\frac{\beta}{\alpha}}(y)f_{\alpha}(s)dyds,$

equivalently,

$\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{PD}(\alpha|s^{-\frac{1}{\alpha}})\mathrm{PD}({\beta}/{\alpha}|y^{-\frac{\alpha}{\beta}})\frac{{\mbox{e}}^{-\lambda
s^{\frac{\beta}{\alpha}}y}}{\mathrm{E}^{(j+1)}_{\beta,j\beta+1}(-\lambda)}g_{\frac{\beta}{\alpha},\frac{j\beta}{\alpha}}(y)g_{\alpha,j\beta}(s)dyds.$

It follows that

$\mathbb{E}\big{[}{\mbox{e}}^{-\lambda
y^{-\frac{\beta}{\alpha}}T^{-\beta}_{\alpha,j\beta}}\big{]}=\sum_{\ell=0}^{\infty}\frac{{(-\lambda
y^{-\frac{\beta}{\alpha}})}^{\ell}}{\ell!}\frac{\Gamma(\frac{\beta(j+\ell)}{\alpha})\Gamma(j\beta)}{\Gamma(\frac{j\beta}{\alpha})\Gamma(j\beta+\ell\beta)}.$
(5.4)

Hence a direct application of Proposition 3.1 leads to identification of all
relevant laws as described in the next result.

######  Proposition 5.1.

Consider the specifications in Proposition 3.1 with
$(\tilde{V},Q)\sim\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(\vartheta^{(\lambda)}_{\beta,j\beta}),$
for each $j=0,1,2\ldots$. Then, in this case,

  1. (i)

$V\sim\mathbb{L}^{(0)}_{\beta,j\beta}(\lambda)$ as in (5.3) with $\theta=0.$

  2. (ii)

$Q$ has the marginal distribution

$\int_{0}^{\infty}\mathrm{PD}(\beta/\alpha|y)\frac{\mathbb{E}\big{[}{\mbox{e}}^{-\lambda
y^{-\frac{\beta}{\alpha}}T^{-\beta}_{\alpha,j\beta}}\big{]}}{{\mathrm{E}^{(j+1)}_{\beta,j\beta+1}(-\lambda)}}f_{\frac{\beta}{\alpha},\frac{j\beta}{\alpha}}(y)dy$

where $\hat{T}_{1}$ has density ${\mathbb{E}\big{[}{\mbox{e}}^{-\lambda
y^{-\frac{\beta}{\alpha}}T^{-\beta}_{\alpha,j\beta}}\big{]}}f_{\frac{\beta}{\alpha},\frac{j\beta}{\alpha}}(y)/{{\mathrm{E}^{(j+1)}_{\beta,j\beta+1}(-\lambda)}},$
which may be expressed in terms of (5.4).

  3. (iii)

$\tilde{V}|\hat{T}_{1}=y,$ has a stable Poisson-Kingman distribution with
index $\alpha$ and mixing density

$\frac{{\mbox{e}}^{-\lambda
s^{-\beta}y^{\frac{-\beta}{\alpha}}}}{\mathbb{E}\big{[}{\mbox{e}}^{-\lambda
y^{-\frac{\beta}{\alpha}}T^{-\beta}_{\alpha,j\beta}}\big{]}}f_{\alpha,j\beta}(s).$

  4. (iv)

$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V)$ has a stable Poisson-Kingman
distribution with index $\alpha$ and mixing density

$\frac{{\mathrm{E}^{(j+1)}_{\frac{\beta}{\alpha},\frac{j\beta}{\alpha}+1}(-\lambda
s^{-\beta})}}{{\mathrm{E}^{(j+1)}_{\beta,j\beta+1}(-\lambda)}}f_{\alpha,j\beta}(s).$

##  6 Coagulation and fragmentation of generalized gamma models

For any $0<\beta<1,$ let $(\tau_{\beta}(s),s\geq 0)$ denote a generalized
gamma subordinator specified by its Laplace transform
$\mathbb{E}[{\mbox{e}}^{-w\tau_{\beta}(s)}]={\mbox{e}}^{-s[(1+w)^{\beta}-1]}.$
The generalized gamma subordinator and corresponding mass partitions, and
bridges defined by normalization, as described in [42], arises in numerous
contexts. However, for purpose of this exposition, the reader may refer to its
role in the construction of $\mathrm{PD}(\beta,\theta)$ distributions as
described in [46, Proposition 21]. More generally, similar to the Mittag-
Leffler class, let $(N(tT_{\beta}(1)),t\geq 0)$ denote a mixed Poisson
process. Then, for $V_{\beta}\sim\mathrm{PD}(\beta,0),$ and for $\zeta>0,$ the
distribution of $V_{\beta}|N(\zeta^{\frac{1}{\beta}}T_{\beta}(1)))=m$
corresponds to the laws
$\mathbb{P}^{[m]}_{\beta}(\zeta):=\mathrm{PK}_{\beta}(r^{[m]}_{\beta,\zeta}\cdot
f_{\beta})$, where

$r^{[m]}_{\beta,\zeta}(t)=\frac{t^{m}{\mbox{e}}^{-\zeta^{\frac{1}{\beta}}t}}{\mathbb{E}[T^{m}_{\beta}{\mbox{e}}^{-\zeta^{\frac{1}{\beta}}T_{\beta}}]},$
(6.1)

for $m=0,1,2,\ldots$, as described in [25, 30, 31]. Here, we show how to use
Proposition 3.1 to easily identify laws and explicit constructions of
(dependent) random measures leading to a coag/frag duality in the case of
$m=0,1$ and also show how one may recover the Poisson-Dirichlet coag/frag
duality results of [41], based on independent $\mathrm{PD}(\alpha,\theta)$ and
$\mathrm{PD}(\frac{\beta}{\alpha},\frac{\theta}{\alpha})$ distributions, in
the case of $\theta>0,$ using $m=0$, and the general case of $\theta>-\beta,$
using $m=1.$ Results for general $m,$ using Proposition 3.1, are also
manageable but require too many additional details for the present exposition.

Now, for a fixed value $\zeta,$ define the scaled subordinator
$\hat{T}_{\beta}(\zeta v)=\tau_{\beta}(\zeta v)/{\zeta^{\frac{1}{\beta}}}$,
for $0\leq v\leq 1,$ such that

$\hat{T}_{1,\beta}(\zeta):=\frac{\tau_{\beta}(\zeta)}{\zeta^{\frac{1}{\beta}}}$

has density
${\mbox{e}}^{-\zeta^{\frac{1}{\beta}}t}{\mbox{e}}^{\zeta}f_{\beta}(t),$ and
one may form a normalized general gamma bridge, for $v\in[0,1],$ as

$F_{P_{\beta}(\zeta)}(v)=\frac{\hat{T}_{\beta}(\zeta
v)}{\hat{T}_{\beta}(\zeta)}=\frac{\tau_{\beta}(\zeta
v)}{\tau_{\beta}(\zeta)}=\sum_{k=1}^{\infty}\hat{P}_{\beta,k}(\zeta)\mathbb{I}_{\\{U_{k}\leq
v\\}},$ (6.2)

where
$P_{\beta}(\zeta):=(\hat{P}_{\beta,k}(\zeta))\sim\mathbb{P}_{\beta}(\zeta):=\mathrm{PK}_{\beta}(r_{\beta,\zeta}\cdot
f_{\beta})$ with
$h(t)=r_{\beta,\zeta}(t):=r^{[0]}_{\beta,\zeta}(t)={\mbox{e}}^{-\zeta^{\frac{1}{\beta}}t}{\mbox{e}}^{\zeta},$
and $\hat{T}_{1,\beta}(\zeta)$ equates to its inverse local time at $1.$ Using
this and Proposition 3.1, for
$V\overset{d}{=}P_{\beta}(\zeta)\sim\mathbb{P}_{\beta}(\zeta),$ we obtain
versions of
$(\tilde{V},Q)\sim\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}(r_{\beta,\zeta})$
in this case as follows; for $Q,$

$h_{\alpha}(y)=\mathbb{E}_{\alpha}\left[r_{\beta,\zeta}(T_{\alpha}y^{\frac{1}{\alpha}})\right]=r_{\frac{\beta}{\alpha},\zeta}(y)={\mbox{e}}^{-\zeta^{\frac{\alpha}{\beta}}y}{\mbox{e}}^{\zeta},$

that is,
$Q\overset{d}{=}P_{\frac{\beta}{\alpha}}(\zeta)\sim\mathbb{P}_{\frac{\beta}{\alpha}}(\zeta)$
with inverse local time at $1,$
$\hat{T}_{1}\overset{d}{=}\hat{T}_{1,\frac{\beta}{\alpha}}(\zeta)$, with
density
${\mbox{e}}^{-\zeta^{\frac{\alpha}{\beta}}y}{\mbox{e}}^{\zeta}f_{\frac{\beta}{\alpha}}(y).$
Hence,
$\tilde{V}|\hat{T}_{1}=y\sim\mathbb{P}_{\alpha}(\zeta^{\frac{\alpha}{\beta}}y),$
and has a marginal distribution, corresponding to
$\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V),$

$\tilde{V}\overset{d}{=}P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta))\sim\mathbb{E}[\mathbb{P}_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta))]:=\int_{0}^{\infty}\mathbb{P}_{\alpha}(\zeta^{\frac{\alpha}{\beta}}y){\mbox{e}}^{-\zeta^{\frac{\alpha}{\beta}}y}{\mbox{e}}^{\zeta}f_{\frac{\beta}{\alpha}}(y)dy.$

Hence, jointly and component-wise,
$(\tilde{V},Q)\overset{d}{=}\left(P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta)),P_{\frac{\beta}{\alpha}}(\zeta)\right),$
and $V\overset{d}{=}P_{{\beta}}(\zeta)$ is determined by the coagulation

$F_{P_{\beta}(\zeta)}(v)=F_{P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta))}\left(F_{P_{\frac{\beta}{\alpha}}(\zeta)}(v)\right),$

where, for clarity,

$F_{P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta))}(v)=\frac{\tau_{\beta}\left(\tau_{\frac{\beta}{\alpha}}(\zeta)v\right)}{\tau_{\beta}\left(\tau_{\frac{\beta}{\alpha}}(\zeta)\right)}\qquad{\mbox{
and }}\qquad
F_{P_{\frac{\beta}{\alpha}}(\zeta)}(v)=\frac{\tau_{\frac{\beta}{\alpha}}(\zeta
v)}{\tau_{\frac{\beta}{\alpha}}(\zeta)}.$

Conversely, $\tilde{V}=\mathrm{FRAG}_{\alpha,-\beta}(V)$ is equivalent in
distribution to
$P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta))=\mathrm{FRAG}_{\alpha,-\beta}(P_{\beta}(\zeta))\sim\mathbb{E}[\mathbb{P}_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\zeta))].$
Now, following [46, Proposition 21], for $\theta>0$, it follows that for
$\gamma_{a}\sim\mathrm{Gamma}(a,1),$
$P_{\beta}(\gamma_{\frac{\theta}{\beta}})\sim\mathrm{PD}(\beta,\theta)$, and

$P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\gamma_{\frac{\theta}{\beta}}))\sim\mathrm{PD}(\alpha,\theta),\quad{\mbox{
independent of }}\quad
P_{\frac{\beta}{\alpha}}(\gamma_{\frac{\theta}{\beta}})\sim\mathrm{PD}\left(\frac{\beta}{\alpha},\frac{\theta}{\alpha}\right),$

where also
$P_{\alpha}(\tau_{\frac{\beta}{\alpha}}(\gamma_{\frac{\theta}{\beta}}))=\mathrm{FRAG}_{\alpha,-\beta}(P_{\beta}(\gamma_{\frac{\theta}{\beta}}))\sim\mathrm{PD}(\alpha,\theta),$
recovering the coag/frag duality in [41] for $\theta>0.$

######  Remark 6.1.

See [29] for an earlier, less refined, treatment of these results which
requires considerably more effort.

###  6.1 Results for size biased generalized gamma
$\mathbb{P}^{[1]}_{\beta}(\zeta)$

In order to recover the duality for the entire range of $\theta>-\beta$, we
now work with the size biased law of a generalized gamma density. Suppose that
$V\overset{d}{=}P^{[1]}_{\beta}(\zeta)\sim\mathbb{P}^{[1]}_{\beta}(\zeta):=\mathrm{PK}_{\beta}\left(r^{[1]}_{\beta,\zeta}\cdot
f_{\beta}\right)$, where

$r^{[1]}_{\beta,\zeta}(t)={\zeta^{\frac{1}{\beta}-1}}t{\mbox{e}}^{-\zeta^{\frac{1}{\beta}}t}{\mbox{e}}^{\zeta}/{\beta},$
(6.3)

and, now

$\hat{T}^{[1]}_{1,\beta}(\zeta):=\frac{\tau_{\beta}\left(\zeta+\gamma_{\frac{1-\beta}{\beta}}\right)}{\zeta^{\frac{1}{\beta}}},$

with density $r^{[1]}_{\beta,\zeta}(t)f_{\beta}(t)$, is the corresponding
inverse local time at $1.$ Since this case and a derivation for
$\theta>-\beta$ is not well known, we apply Proposition 3.1 to identify all
the relevant distributions in the next result.

######  Proposition 6.1.

Consider the variables $V$ and $(\tilde{V},Q),$ forming the coagulation and
fragmentation operations as described in Proposition 3.1, where
$V\overset{d}{=}P^{[1]}_{\beta}(\zeta)\sim\mathbb{P}^{[1]}_{\beta}(\zeta):=\mathrm{PK}_{\beta}\left(r^{[1]}_{\beta,\zeta}\cdot
f_{\beta}\right)$, and thus
$(\tilde{V},Q)\sim\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}\left(r^{[1]}_{\beta,\zeta}\right).$
Then,

  1. (i)

$Q\overset{d}{=}P^{[1]}_{\frac{\beta}{\alpha}}(\zeta)\sim\mathbb{P}^{[1]}_{\frac{\beta}{\alpha}}(\zeta),$
with $\hat{T}_{1}\overset{d}{=}T^{[1]}_{1,\frac{\beta}{\alpha}}(\zeta)$

  2. (ii)

$\tilde{V}|\hat{T}_{1}=y\sim\mathbb{P}^{[1]}_{\alpha}(\zeta^{\frac{\alpha}{\beta}}y)$

  3. (iii)

$\tilde{V}\overset{d}{=}P^{[1]}_{\alpha}\left(\tau_{\frac{\beta}{\alpha}}(\zeta+\gamma_{\frac{\alpha-\beta}{\beta}})\right)$

  4. (iv)

$(\tilde{V},Q)\overset{d}{=}\left(P^{[1]}_{\alpha}\bigg{(}\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}\bigg{)},P^{[1]}_{\frac{\beta}{\alpha}}(\zeta)\right)$
jointly and component-wise.

  5. (v)

$P^{[1]}_{\alpha}\bigg{(}\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}\bigg{)}=\mathrm{FRAG}_{\alpha,-\beta}\left(P^{[1]}_{\beta}(\zeta)\right)$

  6. (vi)

$P^{[1]}_{\beta}(\gamma_{\frac{\theta+\beta}{\beta}})\sim\mathrm{PD}(\beta,\theta)$
for $\theta>-\beta$

  7. (vii)

$P^{[1]}_{\alpha}\left(\tau_{\frac{\beta}{\alpha}}\big{(}\gamma_{\frac{\theta+\beta}{\beta}}+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}\right)\sim\mathrm{PD}(\alpha,\theta)$
independent of
$P^{[1]}_{\frac{\beta}{\alpha}}(\gamma_{\frac{\theta+\beta}{\beta}})\sim\mathrm{PD}(\frac{\beta}{\alpha},\frac{\theta}{\alpha})$

###### Proof.

The results follow from a straightforward application of Proposition 3.1 using
$h(t)=r^{[1]}_{\beta,\zeta}(t),$ the distributional representation of
$\hat{T}_{1}$, and the appropriate Gamma randomization to obtain independent
$\mathrm{PD}$ laws. The generalized gamma subordinator representation of
$\hat{T}_{1}$ and Poisson Dirichlet distributional identities can be found in
[25, 30, 31]. The independence of the Poisson-Dirichlet laws is due to [46,
Proposition 21, see p. 877] and beta-gamma algebra, see also [25, Proposition
2.1]. ∎

$P^{[1]}_{\alpha}\bigg{(}\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}\bigg{)}=\mathrm{FRAG}_{\alpha,-\beta}\left(P^{[1]}_{\beta}(\zeta)\right)$
has distribution

$\mathbb{E}\left[\mathbb{P}^{[1]}_{\alpha}\bigg{(}\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}\bigg{)}\right]$

with inverse local time at $1,$ equivalent in distribution to

$\hat{T}^{[1]}_{1,\alpha}\bigg{(}\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}\bigg{)}\overset{d}{=}\frac{\tau_{\alpha}\left(\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}+\gamma_{\frac{1-\alpha}{\alpha}}\right)}{{[\tau_{\frac{\beta}{\alpha}}\big{(}\zeta+\gamma_{\frac{\alpha-\beta}{\beta}}\big{)}]}^{\frac{1}{\alpha}}}.$

######  Remark 6.2.

If $V_{\alpha}\sim\mathrm{PD}(\alpha,0)$ independent of
$Q_{\frac{\beta}{\alpha}}\sim\mathrm{PD}(\frac{\beta}{\alpha},0)$, it is
evident that
$\big{(}V_{\alpha},Q_{\frac{\beta}{\alpha}}\big{)}\left|N\left(\zeta^{\frac{1}{\beta}}T_{\alpha}\big{(}T_{\frac{\beta}{\alpha}}(1)\big{)}\right)\right.=m\sim\mathrm{P}^{\frac{\beta}{\alpha}}_{\alpha}\left(r^{[m]}_{\beta,\zeta}\right).$

##

The $\mathrm{EPPF}$ of the $\mathrm{PD}_{\alpha|\beta}(\alpha|y)$ Gibbs
partition of $[n]$ in Theorem 4.1 may be alternatively expressed in terms of
Fox-$H$ functions [37] as

$\frac{\alpha
H_{2,2}^{0,2}\left[y\left|\begin{array}[]{l}\left(1-\frac{1}{\beta},\frac{1}{\beta}\right),\left(1-\frac{1}{\alpha}-k,\frac{1}{\alpha}\right)\vspace*{.05in}\\\
\left(1-\frac{1}{\alpha},\frac{1}{\alpha}\right),(-n,1)\end{array}\right.\right]}{H_{1,1}^{0,1}\left[y\left|\begin{array}[]{l}\left(1-\frac{1}{\beta},\frac{1}{\beta}\right)\vspace*{.05in}\\\
\left(0,1\right)\end{array}\right.\right]}\frac{\Gamma(n)}{\Gamma(k)}p_{\alpha}(n_{1},\ldots,n_{k}).$

The above expression follows by noting the Fox-$H$ representations for
$f_{\beta/\alpha}$ and $f^{(n-k\alpha)}_{\alpha,k\alpha},$ followed by
applying [11, Theorem 4.1]. Otherwise details are similar to the arguments in
[24].

L.F. James was supported in part by grants RGC-GRF 16301521, 16300217 and
601712 of the Research Grants Council (RGC) of the Hong Kong SAR.

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