\documentstyle{amsppt}

\magnification=1200
\hcorrection{.25in}
\advance\vsize-.75in

\def\de{\delta}
\def\De{\Delta}
\def\th{\theta}
\def\eps{\varepsilon}
\def\si{\sigma}
\def\Si{\Sigma}
\def\al{\alpha}
\def\Ga{\Gamma}
\def\ga{\gamma}
\def\La{\Lambda}
\def\la{\lambda}
\def\ti{\tilde}
\def\kappa{\varkappa}
\def\be{\beta}
\def\F{{\Cal F}}
\def\Dir{\operatorname{Dir}}

\document
\topmatter
\title {A new proof of the Markov--Krein identity for distributions of
means of Dirichlet processes}\endtitle
\rightheadtext{Markov--Krein identity}
\author N.~Tsilevich and A.~Vershik and M.~Yor
\endauthor
\address N.~Tsilevich, Department of Mathematics and Mechanics,
St.~Petersburg State University, St.~Petersburg, Russia
\endaddress
\email natalia\@pdmi.ras.ru \endemail
\address A.~Vershik,
Steklov Mathematical Institute at St.~Petersburg,
27 Fontanka, 191011 St.~Petersburg, Russia
\endaddress
\email vershik\@pdmi.ras.ru \endemail
\address M.~Yor,
Laboratoire de Probabilit\'es et Mod\`eles Al\'eatoires, Tour 56,
4 Place Jussieu,
75252 Paris Cedex 05, France
\endaddress
\thanks{The part of this work was done when the second author stayed at MSRI.
The first and the second authors are partially supported by
RFBR grant 99--01--00098, and the second author --- by
CRDF grant RM1-2244}
\endthanks
\abstract We present a new simple proof of the Markov--Krein identity
for distributions of means of Dirichlet processes. Our approach is based
on a close relation of Dirichlet processes to gamma processes and 
basic properties of the gamma processes.
\endabstract
\endtopmatter


\subhead{1. Introduction}
\endsubhead
The Dirichlet processes introduced in \cite{F} play a key role
in Bayesian nonparametric statistics. The classical definition of these
processes is as follows. 
Denote by
$\De_n=\{x=(x_0,\ldots,x_n): x_i\ge0,\sum x_i=1\}$ the $n$-dimensional
simplex. The Dirichlet distribution $\Dir(\tau_0,\ldots,\tau_n)$ on $\De_n$
with parameters $\tau_0,\ldots,\tau_n>0$ is determined by the density
$
\frac{\Ga(\tau_0+\ldots+\tau_n)}{\Ga(\tau_0)\ldots\Ga(\tau_n)}
x_0^{\tau_0-1}\ldots x_n^{\tau_n-1}
$.

\definition{Definition 1}
Let $(X,\nu)$ be a standard Borel space with a non-atomic
finite positive measure $\nu$. 
The Dirichlet process on the space $X$ with parameter
measure $\nu$ is a random probability distribution $P$ on $X$ such that
for every finite measurable partition $X=A_0\cup\ldots\cup A_n$, the vector
$(P(A_0),\ldots,P(A_n))$ has Dirichlet distribution
$\Dir(\nu(A_0),\ldots,\nu(A_n))$ on $\De_n$.
\enddefinition

Let $\th=\nu(X)$ be the total charge of the measure $\nu$, and denote by
$\bar\nu=\nu/\th$ the normalized measure $\nu$.
An explicit construction of the Dirichlet process is given by the following
formula:
$$
P=\sum_{i=1}^\infty Q_i\de_{Y_i},
\tag 1
$$
where $Y=(Y_1,Y_2,\ldots)$ is a sequence of of i.i.d.~variables with common
distribution $\bar\nu$, and $Q=(Q_1,Q_2,\ldots)$ is a random point of
the infinite-dimensional simplex
$\De=\{x=(x_1,x_2,\ldots):\,x_1\ge x_2\ge\ldots\ge0, \sum x_i=1\}$
which is independent of $Y$ and has the {\it Poisson--Dirichlet
distribution} $PD(\th)$ with parameter $\th$.

Many papers are devoted to a particular problem of
studying the distributions of
random means of Dirichlet processes.
Let $a:X\to\Bbb R$ be a measurable function.
It defines a linear functional $f_a(\xi)=\int_Xa(x)d\xi(x)$ on
the space of Borel measures on $X$.
The problem is to describe the distribution $\mu_a$
of this functional with respect
to the Dirichlet process $P$. 
The answer is as follows.
Let $\nu_a$ be the distribution of the function $a$ with respect to the
normalized measure $\bar\nu$. Then
the measures $\mu_a$ and $\nu_a$ are related by the following integral identity:
$$
\int_{\Bbb R}\frac1{(1+zu)^\th}d\mu_a(u)=
\exp\left(-\int_X\log(1+zu)^\th d\nu_a(u)\right).
\tag $*$
$$

This formula was first obtained in~\cite{CR} 
by hard analytic arguments (see also simpler proofs 
in~\cite{DK, KT, RGN, HO}). 
In case of $\th=1$, this identity means that
the distribution $\mu_a$ is the {\it Markov--Krein transform} of
the measure $\nu_a$. This transform arises in many contexts, such as
the Markov moment problem, continued fractions theory, 
exponential representations of
functions of negative imaginary type, the spectral shift function
of a self-adjoint operator,
the Plancherel growth of Young diagrams, 
etc.~(see~\cite{Ke} for a detailed survey).
It appeared first in A.~A.~Markov's paper \cite{M} and was intensively
studied by M.~G.~Krein and his school. In particular, this identity
gives a link between the so-called Markov moment problem 
and the well-studied Hausdorff moment problem, 
see \cite{KN}. 

The purpose of this paper is to
present a new proof of the Markov--Krein  identity for
means of Dirichlet processes based
on a relation of Dirichlet processes
to gamma processes. Let $\ga(x)$ be the gamma process on $(X,\nu)$.
Then {\it the Dirichlet process on $X$ with parameter measure $\nu$
is just the normalized gamma process $\bar\ga(x)=\ga(x)/\ga(X)$}.
Thus the above problem about the distributions of means of
Dirichlet processes
may be formulated in terms of gamma processes
as follows: {\it given a linear functional $f_a(\xi)=\int_Xa(x)d\xi(x)$ defined
by a function $a:X\to\Bbb R_+$, describe the
distribution $\mu_a$
of the functional $f_a$ 
with respect to the normalized gamma process $\bar\ga$
in terms of the distribution
$\nu_a$ of $a$}. The key independence property of the gamma processes
(the normalized gamma process $\bar\ga$
is independent on the total charge $\ga(X)$) 
which leads to many
distinguished properties of the gamma processes, e.g.~the
so-called multiplicative quasi-invariance, see \cite{TV,TVY2}, also
allows one to obtain easily
the Markov--Krein formula~($*$) relating these two distributions. This
formula may also be interpreted as a relation 
between the Laplace transform of the distribution
of the functional $f_a$ with respect to the gamma process and the
Cauchy transform of the distribution of the same functional with respect
to the normalized gamma process.

Our interpretation of the Markov--Krein identity leads to the following
general problem: given an arbitrary L\'evy process $\eta$ on $(X,\nu)$
and a function $a:X\to\Bbb R_+$, describe the relation between the
distribution of the functional $f_a$ with respect to $\eta$ and the
distribution of the same functional with respect to the
normalized L\'evy process $\bar\eta$.

\subhead{2. General L\'evy processes}
\endsubhead
It is natural to consider Dirichlet processes
in the context of general L\'evy processes. We present this  framework
following \cite{TV, TVY2}.

Let $(X, \nu)$ be a standard Borel space with a non-atomic finite  
non-negative measure $\nu$, and let $\nu(X)=\th$ be the total charge of $\nu$. 
Denote by 
$$
D=\left\{\sum z_i\de_{x_i},\;x_i\in X,\,z_i\in\Bbb R,\sum|z_i|<\infty \right\}
$$
the real linear space of all finite real discrete measures
on $X$, and by
$D^+=\{\sum z_i\de_{x_i}\in D:\;z_i>0\}\subset D$ the cone in $D$ consisting
of all positive measures. 

Let $\La$ be a measure on $\Bbb R_+$ satisfying the following conditions:
$$
\aligned
&\La(0,\infty)=\infty, \quad\quad\quad\La(1,\infty)<\infty, \\
&\int_0^1sd\La(s)<\infty,  \quad\quad\La(\{0\})=0.
\endaligned
\tag 2
$$
Denote by $\psi_\La$ the Laplace transform of the infinitely divisible
distribution $F_\La$ with L\'evy measure $\La$:
$$
\psi_\La(t)=\exp\left(-\int_0^\infty(1-e^{-ts})d\La(s)\right).
$$

Each bounded Borel function $a:X\to\Bbb R$ defines a linear functional $f_a$
on $D$, where $f_a(\eta)=\int_X a(x)d\eta(x)$ for $\eta\in D$.

\definition{Definition 2}
A L\'evy process on the space $(X,\nu)$ with
L\'evy measure $\La$ satisfying~(2)
is a generalized process on $D$ whose law $P_\La$ has
Laplace transform
$$
\Bbb E\left[\exp\left(-\int_Xa(x)d\eta(x)\right)\right]=
\exp\left(\int_X\log\psi_\La(a(x))d\nu(x)\right),
\tag 3
$$
where $a$ is an arbitrary non-negative bounded Borel function on $X$.
\enddefinition

Consider the cone 
$
C=\{z=(z_1,z_2,\ldots):\;z_1\ge z_2\ge\ldots\ge0,\,\sum z_i<\infty\}\subset l^1
$,
and define a map $T:D^+\to C\times X^\infty$ by
$$
T\eta=\big((Q_1,Q_2,\ldots),\;(X_1,X_2,\ldots)\big),\quad\text{ if }\quad
\eta=\sum Q_i\de_{X_i}.
$$

\definition{Definition 3}
Let $P$ be a distribution on the space $D^+$, and
let $\eta$ be a random process obeying $P$.
The random sequence of charges $(Q_1,Q_2,\ldots)$ is called the
{\it conic part} of the process $\eta$, and its distribution
on the cone $C$ is called
the {\it conic part} of the law $P$.
\enddefinition

It is not difficult to show that the conic part of the L\'evy process
with L\'evy measure $\La$ is the ordered sequence of
points of the Poisson process
on $\Bbb R_+$ with mean measure $|\nu|\La$.
Thus the conic part depends only on $\La$ and on the full charge of
the parameter measure $\nu$.
In fact, the following theorem shows that studying a L\'evy process
may be essentially reduced
to studying its conic part, since the construction of the process involves
the parameter measure in a trivial way.
This fundamental property of L\'evy processes
is a particular case of the representation theorem first
proved in \cite{FK}. A simpler proof of this fact is presented
in~\cite{TVY1}.

\proclaim{Theorem 1 {\rm(\cite{FK, TVY1})}}
Let $\eta=\sum Q_i\de_{X_i}$ be a L\'evy process on the space
$(X,\nu)$ with L\'evy measure $\La$. 
Then $TP_\La=\varkappa_{|\nu|\La}\times\bar\nu^\infty$,
i.e.~$X_1,X_2,\ldots$ is a sequence of i.i.d.~random variables
with common distribution $\bar\nu=\nu/|\nu|$, and this sequence is independent of
the conic part
$(Q_1,Q_2,\ldots)$.
\endproclaim

Denote by $D^+_1\subset D^+$ the simplex
of all normalized atomic measures. Then
$D^+ =D^+ _1 \times [0, \infty)$, i.e.~each $\eta \in D^+$ can be represented as
$$
\eta=({\eta}/{\eta(X)}, \eta(X)).
$$
The second coordinate in this decomposition
is the total charge of the measure $\eta$. It follows from the definition
of the L\'evy process that $\eta(X)$ obeys the infinite divisible law $F_\La$
corresponding to the L\'evy measure $\La$.
The first coordinate
is called the {\it normalization} of the measure $\eta$.
In general, the law of a L\'evy process is not a product measure
in this decomposition. It will be a product measure only for gamma processes,
see below. 

Using this decomposition,
consider a map
$T':D^+\to\Bbb R_+\times\Si\times X^\infty$, where
$
\Si=\{y=(y_1,y_2,\ldots):\;y_1\ge y_2\ge\ldots\ge0,\; y_1+y_2+\ldots=1\}
$
is the infinite-dimensional simplex, and
$$
T'\eta=\big(\eta(X),\;(Q_1/\eta(X),Q_2/\eta(X),\ldots),\;
(X_1,X_2,\ldots)\big),\quad\text{if}\quad\eta=\sum Q_i\de_{X_i}.
$$

\definition{Definition 4}
The normalized sequence of charges $(Q_1/\eta(X),Q_2/\eta(X),\ldots)$
is called the {\it simplicial part} of the process and its distribution 
on $\Si$ is called the {\it simplicial part} of the law $P_\La$.
\enddefinition

See \cite{TVY1} for a detailed treatment of general L\'evy prosesses, 
characterization of conic parts, etc.

\subhead{3. The gamma process}
\endsubhead
In this section we summarize basic properties of the gamma processes which we
need to prove the Markov--Krein identity~($*$).

\definition{Definition 5}
The standard gamma process
on the space $(X,\nu)$
is a L\'evy process $\ga$ on $(X,\nu)$ with L\'evy measure
$d\La(z)=z^{-1}dz$, $z>0$. Thus
the law ${\Cal G}$ of the gamma process has Laplace transform
$$
\Bbb E_{{\Cal G}}\left[\exp\left(-\int_X a(x)d\ga(x)\right)\right]=
\exp\left(-\int_X\log\left(1+a(x)\right)d\nu(x)\right).
\tag 4
$$
\enddefinition

It is easy to show that formula~(4) holds for any measurable function 
$a\in\Cal M=\{a:X\to\Bbb R_+:\,\int_X\log(a(x)+1)d\nu(x)<\infty\}$.

Lemma~1 presents the key independence
property of the gamma process. It 
follows immediately from 
the corresponding property of gamma variables:
if $Y$ and $Z$ are independent gamma variables with the same scale parameter,
then the variables $Y+Z$ and $\frac{Y}{Y+Z}$ are independent. 

\proclaim{Lemma 1} 
The total charge $\ga(X)$
of the gamma process and the normalized gamma process
$\bar\ga=\ga/\ga(X)$ are independent.
The distribution of the total charge is the gamma distribution
with shape parameter $\th=|\nu|$.
\endproclaim

As follows from a remarkable result of Lukacs~\cite{L} for gamma
variables, this property is characteristic of the gamma
processes. That is, if $\eta$ is a L\'evy process such that
$\bar\eta$ and $\eta(X)$ are independent, then $\eta$ is a gamma process
(maybe with some scale parameter).

\proclaim{Lemma 2 {\rm (\cite{Ki})}}
The simplicial part of the gamma process with $|\nu|=\th$
is the Poisson--Dirichlet measure $PD(\th)$.
\endproclaim

It is easy to see from~(3) that for every measurable subset $A\subset X$,
the random variable $\ga(A)$ is governed by the gamma distribution with
shape parameter $\nu(A)$, and for every measurable partition $A_0,\ldots,A_n$
of $X$, the
variables $\ga(A_0),\ldots\ga(A_n)$ are independent. Then it follows from
the known property of the Dirichlet distributions that the vector
$(\ga(A_0)/\ga(X),\ldots,\ga(A_n)/\ga(X))$ has Dirichlet distribution
on the simplex with parameters $(\nu(A_0),\ldots,\nu(A_n))$. Thus
{\it the Dirichlet process on the space $X$ with parameter measure $\nu$ is
just the normalized gamma process on $D(X,\nu)$}.
One can also establish
this relation in terms of the explicit
construction~(1). Indeed, in view of Theorem 1 and Lemmas~1 and~2, 
the right-hand side of~(1) presents the normalized gamma process
on $(X,\nu)$.


\subhead{4. The Markov--Krein identity for means of Dirichlet processes}
\endsubhead
In this section we prove the Markov--Krein identity  
for means of Dirichlet processes which
may be interpreted
as a formula relating the distribution of a linear functional with
respect to the gamma process and the distribution of the same functional
with respect to the normalized gamma process. This interpretation allows one to
prove it immediately, using only a formula for the Laplace
transform of the gamma process and the independence property of the
gamma processes.

Given a function $a\in\Cal M$, 
denote by $\mu_a$ the distribution
of the linear functional $\eta\mapsto f_a(\eta)=\int_X a(x)d\eta(x)$
on $D$ with respect to the law $\Cal D$ of the Dirichlet process
on $(X,\nu)$ (i.e., the law
of the normalized gamma process),
and let $\nu_a$ be the distribution of the function $a$
with respect to the normalized parameter measure $\bar\nu$. 

\proclaim{Theorem 2}
The measures $\mu_a$ and $\nu_a$ are related by the following integral identity,
$$
\int_{\Bbb R}\frac1{(1+zu)^\th}d\mu_a(u)=\exp\left(-\int_X\log(1+zu)^\th d\nu_a(u)\right).
\tag 5
$$
\endproclaim

Note that the left-hand side of (5) is the generalized 
Cauchy--Stieltjes transform of the distribution $\mu_a$. It is natural to call
the right-hand side the multiplicative version of the generalized 
Cauchy--Stieltjes transform of the distribution $\nu_a$. In view of~(4), 
it is equal
to the Laplace transform of the gamma process 
calculated on the function $a$.
Hence one may regard formula~(5) as relating an integral transform 
(Cauchy--Stieltjes) of
the distribution $\mu_a$ of the functional $f_a$ with respect to the normalized
gamma process and an integral transform (Laplace) of
its distribution with respect to the non-normalized gamma process. 

\demo{Proof} Using~(4), Lemma~1
and the Fubini theorem we obtain that
the right-hand side of~(5) equals
$$
\align
&\!\!\!\!\!\!\!\!\!\!\exp\left(-\int_X\log (1+za(x))d\nu(x)\right)=
\Bbb E_{\Cal G}\left[\exp\left(-z\int_X a(x)d\ga(x)\right)\right]\\
\qquad=&\Bbb E_{\Cal G}\left[\exp\left(-z\ga(X)\int_X a(x)d\bar\ga(x)\right)\right]\\
\qquad=&\Bbb E_{\Cal D}\left[\frac1{\Ga(\th)}\int_0^\infty t^{\th-1}
\exp\left(-t-zt\int_Xa(x)d\bar\ga(x)\right)\right]\\
\qquad=&\Bbb E_{\Cal D}\left[\frac1{(1+z\int_Xa(x)d\bar\ga(x))^\th}\right],
\endalign
$$
and Theorem follows.
\enddemo         
    
According to a personal communication of P.~Diaconis to S.~Kerov, the idea
of proving formula~(6) using the Laplace transform formula for the gamma
process was used by F.~Huffer in case of {\it discrete} parameter 
measure $\nu$ (in
this case the gamma process is just a sum of independent gamma variables, and
the normalized gamma process is a random point of a finite-dimensional simplex
obeying a Dirichlet distribution). But the fact that this argument works 
as well for continuous parameter measures  
seems to have been overlooked. 
But after the preliminary version \cite{TVY1}
of this paper had been published, other 
independent close 
proofs of the identity~(5) appeared, see \cite{RGN, HO}.

A multivariate version of the Markov--Krein identity for 
the common distributions of
several linear functionals of the Dirichlet process
was first obtained in
\cite{KT}. It is easy to extend our proof
of Theorem~2 to obtain this result.

\proclaim{Theorem 3} Let $a_1,\ldots,a_n\in\Cal M$.
Denote by $\mu_a$ the common
distribution
of the linear functionals $(f_{a_1},\ldots,f_{a_n})$
on $D$ with respect to the law $\Cal D$ of the Dirichlet process.
Let $\nu_a$ be the common distribution of the functions $a_1,\ldots,a_n$
with respect to the normalized parameter measure $\bar\nu$.
Then the
measures $\mu_a$ and $\nu_a$ are related by the 
multivariate Markov--Krein identity
$$
\multline
\int_{\Bbb R^n}\frac1{(1+z_1u_1+\ldots+z_nu_n)^\th}d\mu_a(u)\\=
\exp\left(-\int_{\Bbb R^n}\log(1+z_1u_1+\ldots+z_nu_n)^\th d\nu_a(u)\right).
\endmultline
\tag 6
$$
\endproclaim

\demo{Proof} Goes exactly as the proof of Theorem~2 with function $za(x)$
replaced by $z_1a_1(x)+\ldots+z_na_n(x)$.
\enddemo


\subhead{5. Two-parameter generalization of the Markov--Krein identity}
\endsubhead
S.~Kerov \cite{Ke} and J.~Pitman \cite{P} suggested independently
the same class of generalizations of the Dirichlet processes.
These generalizations are indexed by an arbitrary distribution $\si$ on
the simplex $\De$, and they are defined
by~(1), where the sequence $Q$ obeys
$\si$ instead of $PD(\th)$. 
An important particular case of these generalized
Dirichlet processes is obtained when $\si$ is the so-called two-parameter
Poisson--Dirichlet distribution $PD(\al,\th)$ \cite{PY}.
The range of admissible parameters is the union of the sets
$\{(\al,\th):\,\al\in(0,1),\th>-\al\}$ and
$\{(\al,-m\al):\,\al<0,m\in\Bbb N\}$. When $\al=0$, the measure $PD(0,\th)$
coincides with the ordinary Poisson--Dirichlet distribution $PD(\th)$.
Denote by $\Cal D(\al,\th)$ the law of
the generalized Dirichlet process associated with
the two-parameter Poisson--Dirichlet distribution $PD(\al,\th)$.

An analogue of the Markov--Krein identity for the distribution of a linear
functional with respect to $\Cal D(\al,\th)$
is obtained in \cite{T}.
We present here a new proof of this identity based on relation of the
two-parameter Poisson--Dirichlet distributions to the stable processes.

\definition{Definition 6}
Let $\al\in(0,1)$.
The standard $\al$-stable process
on the space $(X,\nu)$ is
a L\'evy process with L\'evy measure
$d\La_\al=\frac{\al}{\Ga(1-\al)}s^{-\al-1}ds$, $s>0$.
Thus the law $P_\al$ of the $\al$-stable process has
Laplace transform
$$
\Bbb E_{P_\al}\left[\exp\left(-\int_Xa(x)d\eta(x)\right)\right]=
\exp\left(-\int_Xa(x)^\al d\nu(x)\right),
\tag 7
$$
where $a:X\to\Bbb R_+$ is an arbitrary measurable function with
$\int_Xa(x)^\al d\nu(x)<\infty$.
\enddefinition

\proclaim{Lemma 3 \cite{PY}} 
The simplicial part of the $\al$-stable process 
is the Poisson--Dirichlet distribution $PD(\al,0)$.
\endproclaim

The Poisson--Dirichlet distribution $PD(\al,\th)$ with $\al,\th\ne0$
cannot be represented as the simplicial part of any L\'evy process. 
However, one may obtain it
as the simplicial part of the process that has density with respect to a stable
process. Namely,
let $\th>-\al$ and consider the law $P_{\al,\th}$ on $D$ which has density
$$
\frac{dP_{\al,\th}}{dP_\al}(\eta)=\frac{c_{\al,\th}}{\eta(X)^\th}
\tag 8
$$
with respect to the $\al$-stable law $P_\al$. Here 
$c_{\al,\th}=\frac{\Ga(\th+1)}{\Ga(\th/\al+1)}$
is a normalizing constant.

\proclaim{Lemma 4 \cite{PY}}
The simplicial part of the measure $P_{\al,\th}$ is the Poisson--Dirichlet
distribution $PD(\al,\th)$.
\endproclaim

It follows from Lemmas~3  and~4
that the generalized Dirichlet process with parameters $(\al,0)$
is just the normalized $\al$-stable process, and the generalized
Dirichlet process with parameters
$(\al,\th)$ is the normalized processed governed by the law
$P_{\al,\th}$.

Given an arbitrary measurable function $a:X\to\Bbb R_+$ with
$\int_Xa(x)^\al d\nu(x)<\infty$,
let $\mu_a$ be the
distribution of the functional $f_a$ with respect to $\Cal D(\al,\th)$, 
and let $\nu_a$ be the distribution of $a$
with respect to the normalized parameter measure $\bar\nu$. 

\proclaim{Theorem 4} 
The measures $\mu_a$ and $\nu_a$ are related by the following
integral identity:

\noindent {\rm 1)} if $\th\ne0$,
$$
\left(\int_{\Bbb R}(1+zu)^{-\th}d\mu_a(u)\right)^{-\frac1\th}=
\left(\int_{\Bbb R}(1+zu)^\al d\nu_a(u)\right)^\frac1\al;
\tag 9
$$

\noindent {\rm 2)} if $\th=0$,
$$
\exp\left(
\int_{\Bbb R}\log(1+zu)^\al d\mu_a(u)\right)=\int_{\Bbb R}(1+zu)^\al d\nu_a(u).
\tag 10
$$
\endproclaim

\demo{Proof} 1) Denote the left-hand side of the desired identity
by $A^{-1/\th}$ and the right-hand side by $B^{1/\al}$.
Using the identity
$$
\frac1{r^\th}=\frac1{\Ga(\th)}\int_0^\infty t^{\th-1}e^{-rt}dt, 
$$
we obtain
$$
\align
A=&c_{\al,\th}\Bbb E^\al\left[\left(\eta(X)+z\int_Xa(x)d\eta(x)\right)^{-\th}\right]\\
=&\frac{c_{\al,\th}}{\Ga(\th)}\Bbb E^\al\left[
\int_0^\infty t^{\th-1}\exp\left(-t\left(\eta(X)+z\int_Xa(x)d\eta(x)\right)\right)dt
\right]\\
=&\frac{c_{\al,\th}}{\Ga(\th)}\int_0^\infty t^{\th-1}\Bbb E^\al\left[
\exp\left(-\int_Xt(1+za(x))d\eta(x)\right)\right]dt.
\endalign
$$
By the Laplace transform formula~(7), 
the expectation equals precisely $e^{-t^\al B}$, thus
$$
A=\frac{c_{\al,\th}}{\Ga(\th)}\int_0^\infty t^{\th-1}e^{-t^\al B}dt,
$$ 
and~(9) follows by changing variables.

2) Follows from~(9) by letting $\th\to0$. 
\enddemo


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\endRefs

\enddocument