\documentstyle{amsppt}
%\mag=1100
\magnification=\magstep1
\def\publogo{\hfil}
\NoRunningHeads

\define\de{\delta}
\define\th{\theta}
\define\eps{\varepsilon}
\define\si{\sigma}
\define\Si{\Sigma}
\define\al{\alpha}
\define\Ga{\Gamma}
\define\ga{\gamma}
\define\La{\Lambda}
\define\la{\lambda}
\define\ti{\tilde}

\topmatter
\title Quasi-invariance of the gamma process and
multiplicative properties of the Poisson--Dirichlet measures
\endtitle
\author
N.~Tsilevich, A.~Vershik, M.~Yor
\endauthor
\thanks First two authors partially supported by RFBR grant 99-01-00098
\endthanks
\abstract
In this paper we describe new fundamental properties of the law
$P_\Gamma$ of the classical gamma process and related properties of the
Poisson--Dirichlet measures $PD(\theta)$. We prove 
the quasi-invariance of the measure $P_\Gamma$ with respect to an infinite-dimensional 
multiplicative
group (the fact first discovered in \cite{GGV83}) and the Markov--Krein identity
as corollaries of the formula for
the Laplace transform of $P_\Ga$. 

The quasi-invariance of the measure $P_\Ga$ allows us to obtain new
quasi-invariance properties of the measure $PD(\theta)$. 
The corresponding invariance properties hold for 
$\si$-finite analogues of $P_\Ga$ and $PD(\th)$. 
We also show that the measure $P_\Gamma$ can be considered as a
limit of measures corresponding to the $\al$-stable L\'evy processes when parameter
$\alpha$ tends to zero.     

Our approach is based on simultaneous considering the gamma process (especially
its Laplace transform) and its simplicial part -- the Poisson--Dirichlet measures.
\bigskip
\centerline{\bf Quasi-invariance du processus gamma et propri\'et\'es 
multiplicatives}
\centerline{\bf des distributions de Poisson--Dirichlet} 
\bigskip
\noindent{\smc R\'esum\'e.}
Dans cette Note, nous d\'ecrivons certaines propri\'et\'es fondamentales de la loi
$P_\Ga$ du processus gamma classique et des propri\'et\'es correspondantes
des m\'esures de Poisson--Dirichlet $PD(\th)$.
Nous d\'eduisons la propri\'et\'e de quasi-invariance 
de la loi $P_\Ga$ par rapport \`a un ``gros'' groupe multiplicatif 
et l'identit\'e de Markov--Krein directement
de la transformation de Laplace de $P_\Ga$. 

La quasi-invariance de la loi $P_\Ga$ 
permet d'obtenir des propri\'et\'es de quasi-in\-varian\-ce des lois de Poisson--Dirichlet
$PD(\th)$. On obtient les propri\'et\'es correspondantes d'invariance pour
des analogues $\si$-finis de  $P_\Ga$ et $PD(\th)$.
Nous montrons enfin que la loi $P_\Ga$ peut \^etre consider\'ee comme une limite de lois de
processus stables dont le param\`etre $\al$ tend vers $0$. 

Notre approche se fonde sur la consid\'eration simultan\'ee du processus gamma
(surtout sa transformation de Laplace) et sa partie simpliciale -- des m\'esures de
Poisson--Dirichlet.
\endabstract
\endtopmatter
\document

\subhead  1. Introduction: definition of the gamma process 
\endsubhead

In this section 
we present a definition  of the gamma process on an arbitrary space. This general
definition turns out to be  
more convenient for our purposes than the process on the interval.
      
Let $(X, \nu)$ be a standard Borel space with a non-atomic finite  
non-negative measure $\nu$, and let $\nu(X)=\theta$ be the total charge of $\nu$. 
We denote by $D=\{\sum z_i\de_{x_i},\;x_i\in X,\,z_i\in\Bbb R,\sum|z_i|<\infty \}$ 
a real linear space of all finite real atomic measures 
on $X$. 

\smallskip\noindent
{\bf Definition.} {\it The gamma process} on the space $X$ 
with parameter measure $\nu$ is a generalized process on
the space $D$ with the law $P_\Ga=P_\Ga(\nu)$ (called the {\it gamma measure}
on the space $(X,\nu)$)
given by the characteristic functional (Laplace transform) 
$$
\Bbb E_{P_\Ga}\left[\exp\left(-\int_X a(x)d\eta(x)\right)\right]=
\exp\left(-\int_X\log(1+a(x))d\nu(x)\right),
\tag 1.1
$$
where $a$ is an arbitrary non-negative bounded Borel function on the space $X$. 
 
The correctness of this definition is guaranteed by the following explicit construction
(see \cite{Ki93}, chapter~8). Consider a Poisson point process on the space
$X\times\Bbb R_+$ with mean measure $\nu\times\La$, where $\La$ is the L\'evy measure
of the gamma process, that is $d\La(z)=z^{-1}e^{-z}dz$, $z\in\Bbb R^+$.
We associate with a realization $\Pi=\{(x_i,z_i)\}$ of this process an element
$\eta=\sum z_i\de_{x_i}\in D$. Then $\eta$ is a random atomic measure obeying
the law $P_\Ga$. So the gamma measure is concentrated on the cone 
$D^+\subset D$ consisting
of all finite positive atomic measures on X. 

Let $\Cal M=\Cal M(X,\nu)$ be the set of (classes\!$\mod 0$ of)
non-negative measurable functions on the space $X$
with $\nu$-summable logarithm,
$$
\Cal M=\left\{a:X\to\Bbb R_+:\;\int_X|\log a(x)|d\nu(x)<\infty\right\}.
$$
It follows from the above Poisson construction and formula~(1.1) that
each function $a\in\Cal M$ correctly defines a measurable linear functional
$\eta\mapsto f_a(\eta)=\int_X a(x)d\eta(x)$ on $D$, and formula~(1.1) holds for
all $a\in\Cal M$.

Denote by $D^+_1\subset D^+$ the simplex 
of all normalized atomic measures. Then 
$D^+ =D^+ _1 \times [0, \infty)$, that is each $\eta \in D^+$ can be represented as
$$
\eta=({\eta}/{\eta(X)}, \eta(X)). 
\tag 1.2
$$
The second coordinate in this 
decomposition is the total charge of the measure $\eta$, and the first one is called
the {\it normalization} of the measure $\eta$. 

The following lemma presents a well-known independence property of the gamma process. 

\proclaim{Lemma 1} In representation~(1.2) the gamma measure is a product measure
$P_\Gamma = \Cal G_\th\times\bar P_\Gamma$, that is the total charge $\ga(X)$
of the gamma process and the normalized gamma process $\bar\ga=\ga/\ga(X)$ are independent.
The distribution $\Cal G_\th$ of the total charge is the gamma distribution
on $\Bbb R_+$ with shape parameter $\th$ and scale
parameter 1, i.e. $d\Cal G_\th=\frac1{\Ga(\th)}t^{\th-1}e^{-t}dt$, $t>0$.
\endproclaim
     
\smallskip\noindent
{\bf Remarks.}
{\bf 1.} This independence property characterizes the gamma
process in the class of L\'evy processes (see \cite{Lu??}).

{\bf 2.} The random probability measure $\bar\ga=\ga/\ga(X)$ is known in the literature as 
the {\it Dirichlet process} on the space $X$ with parameter measure $\nu$ 
(see~\cite{Fe73}). 

{\bf 3.} Our definition of the gamma process on an arbitrary space is closely related 
to a particular case of the completely random measure considered in~\cite{Ki93}, 
chapter~8.

{\bf 4.} It is clear that the ordinary definition of the gamma
subordinator on $\Bbb R_+$ is obtained for $X=\Bbb R_+$ and $\nu$ equal to
the Lebesgue measure.

\smallskip
    
\subhead 2. Multiplicative quasi-invariance of the gamma process
\endsubhead 
It follows immediately from the formula~(1.1)
that the measure $P_\Gamma(\nu)$ is invariant under all $\nu$-preserving 
transformations of the space $(X,\nu)$. More exactly, let $T:X\to X$ be a $\nu$-preserving
transformation, then 
the operator $U_T$, which acts on $D$ by substituting the coordinates,  
preserves the measure $P_\Gamma$.   
Now we present a large group of linear transformations of the space $D$ 
(preserving the cone $D^+$) for which $P_\Gamma$ is a quasi-invariant measure. 
    
Consider the defined above class $\Cal M$ of non-negative functions on $X$ 
with $\nu$-summable logarithm.
Each function $a\in\Cal M$ defines not only a linear functional $f_a$ on $D$ but also
a multiplicator $M_a:D\to D$ by $(M_a\eta)(x)=a(x)\eta(x)$, that is 
$M_a\eta=\sum a(x_i)z_i\de_{x_i}$ for $\eta=\sum z_i\de_{x_i}$.
Note that the set $\Cal M$ is a commutative group with respect to pointwise
multiplication of functions, and $M_a$ is a group action of $\Cal M$.
Denote by $\ti a$ the function $\ti a(x)=1/a(x)-1$.

\proclaim{Theorem 1}
1) For each $a\in\Cal M$, the measure $P_\Ga$ is
quasi-invariant under $M_a$, and the corresponding density is given by
the following formula,
$$
\frac{d(M_aP_\Ga)}{dP_\Ga}(\eta)=\exp\left(-\int_X\log a(x)d\nu(x)\right)\cdot
\exp\left(-\int_X\ti a(x)d\eta(x)\right).
\tag 2.1
$$

2) The action of the group $\Cal M$ 
on the space $(D^+,P_\Gamma)$  is ergodic.
\endproclaim        

\smallskip\noindent
{\bf Remarks. 1.} This property of the gamma process was first discovered 
in~\cite{GGV83, GGV85} in quite different terms; 
it plays an important role in the representation theory  
of the current group $SL(2,F)$, where $F$ is the space of functions on
a manifold.
        
{\bf 2.} It is natural to ask whether the quasi-invariance
property stated in Theorem~1 is characteristic of the gamma process. 
If we fix the density, then the answer is positive, i.e.
the gamma measure is the only measure on $D^+$ 
satisfying the formula~(2.1). On the other hand,  
if we consider a smaller group of multiplications, then the following example shows that the
answer is negative.
Let us call a process {\it quasi-multiplicative} if its law is invariant 
under all transformations $M_a$ with {\it constant} $a>0$. For simplicity, 
consider {\it subordinators} (that is
L\'evy processes for $X=\Bbb R_+$ and $\nu$ equal to the Lebesgue measure
on $\Bbb R_+$). The following fact was stated in~\cite{VY95}. Let 
$\eta$ be a subordinator
with L\'evy measure $\La(dx)=k(x)dx$, where $k(x)>0$, and denote
$g(x)=xk(x)$. Then $\eta$
is quasi-multiplicative if and only if for all $a>0$
$$
\int_0^1\left(\sqrt{g(x\,/\,a)}-\sqrt{g(x)}\right)^2\frac{dx}x<\infty.
$$
It is shown in \cite{VY95} that for each $m<1\,/\,2$ any function $k_m(x)$ that
satisfies
$$
k_m(x)=\frac1x\left(\log\frac1x\right)^{2m}\text{ for \ \ }x<1\,/\,2
\qquad\text{ and }\qquad
\int_{1\,/\,2}^\infty k_m(x)dx<\infty
$$
provides an example of a quasi-multiplicative subordinator that is not equivalent to any
scaled gamma process.
It is clear that if $\eta$ is quasi-multiplicative,
then its law is quasi-invariant under all step functions
with finitely many steps, but the quasi-invariance 
under the whole group $M$ does not take place in this example.
It is not known if there exists a measure different from $P_\Gamma$
which has this property.
\smallskip

\subhead 3. Decomposition of the laws of general L\'evy processes
\endsubhead

Let $(X,\nu)$ be a standard Borel space, $\nu(X)=\th$ and $\bar\nu=\nu/\th$. 
Consider a measure $\La$ on $\Bbb R_+$ satisfying the following properties,
$$
\La(0,\infty)=\infty,\qquad \La(1,\infty)<\infty,\qquad
\int_0^1sd\La(s)<\infty,\qquad\La(\{0\})=0.
\tag 3.1
$$
Let $F_\La$ be the infinitely divisible
distribution with L\'evy measure $\La$, 
i.e. the Laplace transform $\psi_\La$ of $F_\La$ is given by
$$
\psi_\La(t)=\exp\left(\int_0^\infty(1-e^{-ts})d\La(s)\right).
$$ 
A homogeneous L\'evy process on $(X,\nu)$ with
L\'evy measure $\La$ satisfying~(3.1) is a process on $D^+$ whose law $P_\La=P_\La(\nu)$ has
the Laplace transform
$$
\Bbb E\left[\exp\left(-\int_Xa(x)d\eta(x)\right)\right]=
\exp\left(\int_X\ln\psi_\La(a(x))d\nu(x)\right).
$$
One may obtain this process explicitly via the Poisson process construction similar 
to that for the gamma process (see Section~1).

Let $C=\{y=(y_1,y_2,\ldots):\;y_1\ge y_2\ge\ldots\ge0,\,\sum y_i<\infty\}\subset l^1$ 
be the cone. We define a class of measures on $C$ as follows.
Let $n\in\Bbb N$ and $p_1,\ldots, p_n>0$, $p_1+\ldots+p_n=1$.
Consider a sequence $\xi_i$ of i.i.d. variables such that 
$P(\xi_i=k)=p_k$ for $k=1,\ldots n$. For $(Q_1,Q_2,\ldots)\in C$, denote by $\Si_k$
the sum $\Si_k=\sum_{i:\xi_i=k}Q_i$. Let $\varkappa$ be a measure on the cone $C$ such that 
the distribution $F$ of the sum $\sum Q_i$ with respect to $\varkappa$ is infinitely 
divisible.  
We say that a measure $\varkappa$ is {\it quasi-product}, if for each 
$n\in\Bbb N$ and each sequence   $p_1,\ldots, p_n>0$, $p_1+\ldots+p_n=1$,
the sums $\Si_1,\ldots\Si_n$ are independent and $\Si_k$ obeys the law
$F^{*p_k}$. 
Note that the ordering and the positiveness
of sequences do not matter in this definition, so it
applies to the whole space $l^1$.
In case of distributions concentrated on sequences
with a bounded number of elements, a quasi-product measure is just an ordinary 
product measure. 
Note also that a quasi-product measure 
on the cone is defined by just one distribution on the half-line.

We define a map $T:D^+\to C\times X^\infty$ by
$$
T\eta=\big((Q_1,Q_2,\ldots),\;(X_1,X_2,\ldots)\big),\quad\text{ if }\quad
\eta=\sum Q_i\de_{X_i}.
$$ 
Let $P$ be a distribution on the space $D^+$, and 
let $\eta$ be a random process obeying the law $P$. Then
the random sequence of charges $Q_1,Q_2,\ldots$ is called the
{\it conic part} of the process, and its distribution  
on the cone $C$ is called
the {\it conic part} of the law $P$.
 
The first part of the following theorem is a 
fundamental property of homogeneous L\'evy processes
that was first proved in \cite{FK72}. We will present a simpler proof in the detailed
version of this paper. The second part of the theorem seems to be new.

\proclaim{Theorem 2} 
1) Let $\eta=\sum Q_i\de_{X_i}$ be a homogeneous L\'evy process on the space 
$(X,\nu)$ with L\'evy measure $\La$. 
Then $TP_\La=\varkappa_\La\times\bar\nu^\infty$, that is 
$X_1,X_2,\ldots$ is a sequence of i.i.d. random variables
with common distribution $\bar\nu$, and this sequence is independent of 
the conic part
$\{Q_i\}_{i\in\Bbb N}$.  

2) The measure $\varkappa$ on the cone $C$ is a conic part of some L\'evy process
$P_\La$ if and only if it is quasi-product with $F=F_\La$. 
\endproclaim        

One may also 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
$$
T'\eta=\big(\eta(X),\;(Q_1/\eta(X),Q_2/\eta(X),\ldots),\;
(X_1,X_2,\ldots)\big),\quad\text{ if }\quad\eta=\sum Q_i\de_{X_i}.
$$
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 $\si_\La$
is called the {\it simplicial part} of the law $P_\La$.

\proclaim{Theorem 3}
The simplicial part of the gamma 
measure $P_\Ga(\nu)$ with $\nu(X)=\th$ is the {\it Poisson--Dirichlet}
distribution $PD(\th)$ with parameter $\th$.
\endproclaim 

This fact was mentioned in~\cite{Ki75} but it seems that the advantages of 
simultaneous studying both measures $P_\Ga$ and $PD(\th)$ were not used systematically 
before.

It follows from Lemma~1 that the conic part of the gamma measure is
a product measure  $\Cal G_\th\times PD(\th)$, where $\Cal G_\th$ is the gamma 
distribution on
$\Bbb R_+$ with parameter $\th$, that is
$T'P_\Ga=\Cal G_\th\times PD(\th)\times\bar\nu^\infty$. This fact, characterizing
the gamma process in the class of L\'evy processes, is a key point for many 
important properties of $P_\Ga$. 

The simplicial part of the stable L\'evy process with 
parameter $\alpha$ is the measure
$PD(\alpha,0)$ on the simplex $\Sigma$  
as defined in \cite{PY97}. This is the most natural 
definition of these measures. Some modification of this construction gives
the whole two-parameter family $PD(\al,\th)$ from \cite{PY97}. 

\subhead 4. Equivalent $\sigma$-finite extensions of the measures $P_\Ga$ and $PD(\th)$
\endsubhead         
         
In this section we define, following \cite{GGV83, GGV85},
a $\si$-finite measure on $D^+$ which is equivalent to $P_\Ga$ and 
invariant under $M_a$ for all functions $a\in\Cal M$ such that
$\int_X\log a(x)d\nu(x)=1$. Namely,
consider a $\si$-finite measure $\ti P_\Ga$ on $D^+$ defined by 
$$
\frac{d\ti P_\Ga}{dP_\Ga}(\eta)=\exp(\eta(X)).
$$

Obviously, 
$T'\ti P_\Ga=m_\th\times PD(\th)\times\bar\nu^\infty$ and
$T\ti P_\Ga=\widetilde{PD}(\th)\times\bar\nu$, 
where $m_\th$ has density $t^{\th-1}\,/\,\Ga(\th)$, $t>0$
(in particular, $m_1$ is just the Lebesgue
measure on the half-line), 
and $\widetilde{PD}(\th)=m_\th\times PD(\th)$.

Theorem 1 implies 

\proclaim{Theorem 4}
For each $a\in\Cal M$, the measure $\ti P_\Ga$ is quasi-invariant under
$M_a$ with a constant density
$$
\frac{dM_a(\ti P_\Ga)}{d\ti P_\Ga}=\exp\left(-\int_X\log a(x)d\nu(x)\right).
$$
\endproclaim

\proclaim{Corollary}
If $\int_X \log a(x)d\nu(x)=0$, then $\ti P_\Gamma$ is invariant
with respect to $M_a$. 
\endproclaim

We see that the measure  $\ti P_\Gamma$ is invariant with respect to an 
infinite-dimensional multiplicative group whose action is like a continual analogue 
of the action of the group of diagonal matrices with determinant $1$ in
a finite-dimensional vector space, so we can consider the measure 
$\ti P_\Gamma$ as an infinite-dimensional analogue of the Lebesgue measure.
This property was much used in 
\cite{GGV83, GGV85} for the representation theory 
of the group $SL(2,F)$. 

\subhead 5. Quasi-invariance of the Poisson--Dirichlet distributions
\endsubhead

Let $a\in\Cal M$.
According to the general theory of polymorphisms (see \cite{Ve77}),
the transformation $M_a$ induces a Markovian operator $R_{a}$ on the cone $C$.
Namely, let $y=(y_1,y_2,\ldots)\in C$. Consider the conditional
distribution $\bar P^y_\Ga$ of the gamma process on $(X,\nu)$, 
given the conic part equal
$(y_1,y_2,\ldots)$. Then the random image of the point $y$ under
$R_{a}$ is the conic part
of the process $M_a\eta$, where $\eta$ 
obeys the law $\bar P_\Ga^y$. It follows from Theorem~2 that 
$$
R_{a}y=V({a(X_1)y_1},{a(X_2)y_2},\ldots),
$$
where $(X_1,X_2,\ldots)$ is a sequence of i.i.d. random variables on $X$
with common distribution $\bar\nu$,
and  $V$ denotes a map that
arranges the coordinates in non-increasing order.

In a similar way, the transformation $M_a$ induces
a Markovian operator $S_{a}$ on the simplex $\Si$,
$$
S_{a}y=V\left(\frac{a(X_1)y_1}{\si},\frac{a(X_2)y_2}{\si},\ldots
\right),
$$
where the sequence $(X_1,X_2,\ldots)$ is as before, and 
$\si=a(X_1)y_1+a(X_2)y_2+\ldots$.

Note that the definitions of the operators $S_a$ and $R_a$ depend only on 
the distribution of the function $a$. Thus when
studying the Poisson--Dirichlet distributions we may assume that $X=[0,1]$ and
$\nu=\th\la$, where $\la$ is the Lebesgue measure on the interval.

Theorems~1 and~4  immediately imply

\proclaim{Theorem 5}
1) The Poisson--Dirichlet distribution $PD(\th)$
is quasi-invariant under $S_{a}$
for all $a\in\Cal M$, and 
$$
\frac{dS_{a}PD(\th)}{dPD(\th)}(y)=
\exp\left(-\th\int_0^1\log a(s)ds\right)\cdot
\int_0^\infty\frac{\si^{\th-1}}{\Ga(\th)}
\left(\prod_{i=1}^\infty F_{1/a}(\si y_i)\right)d\si,
$$
where $F_{1/a}(\cdot)$ is the Laplace transform of the distribution
of the function $1/a(t)$ with respect to the uniform
distribution on the interval $[0,1]$.

2) The Poisson--Dirichlet distribution $PD(\th)$ is
ergodic with respect to $\{S_{a}\}_{a\in\Cal M}$.

3) The $\si$-finite measure $\widetilde{PD}(\th)$ on the cone $C$
is invariant under $R_{a}$
for all $a\in\Cal M$.
\endproclaim

\subhead 6. Distributions of linear functionals of the gamma processes and the
Markov--Krein transform
\endsubhead        

In this section we relate the distribution of a function $a\in\Cal M$ with
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 $\bar P_\Ga$ of the normalized gamma process 
using only the formula for the Laplace transform of $P_\Ga$.
More exactly, denote by $\mu_a$ the
distribution of the functional $f_a$ with respect to $\bar P_\Ga$, 
and let $\nu_a$ be the distribution of the function $a$
with respect to the normalized measure $\bar\nu$. 
The following property is 
characteristic for the gamma process. 

\proclaim{Theorem 6} 
The measures $\mu_a$ and $\nu_a$ are related by the following integral identity,
$$
\int_{\Bbb R}\frac1{(1+zu)^\th}d\mu_a(u)=\exp\left(-\int_X\log(1+zu)^\th d\nu_a(u)\right).
\tag 6.1
$$
\endproclaim

This formula was first obtained in~\cite{CR90} in the context of Dirichlet processes
by hard analytic arguments (see also simpler proofs in~\cite{DK96} 
and~\cite{KTs98}). But the relation with the gamma process, making the proof very
simple, so far has escaped one's attention. 
Note that the left hand side of the identity~(6.1) 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~(1.1), it is equal
to the Laplace transform of the gamma measure $P_\Ga$ calculated on the function $a$.

In case of $\th=1$, the identity~(6.1) 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 Plancherel growth of Young diagrams, etc.
(see~\cite{Ke98} for a detailed survey).

\demo{Proof of Theorem 6} Using~(1.1), Lemma~1 and the Fubini theorem we obtain that 
the right hand side of~(6.1) equals
$$
\align
\exp&\left(-\int_X\log (1+za(x))d\nu(x)\right)=
\Bbb E_{P_\Ga}\left[\exp\left(-z\int_X a(x)d\ga(x)\right)\right]\\
&\qquad=\Bbb E_{P_\Ga}\left[\exp\left(-z\ga(X)\int_X a(x)d\bar\ga(x)\right)\right]\\
&\qquad=\Bbb E_{P_{\bar\Ga}}\left[\frac1{\Ga(\th)}\int_0^\infty t^{\th-1}
\exp\left(-t-zt\int_Xa(x)d\bar\ga(x)\right)\right]\\
&\qquad=\Bbb E_{P_{\bar\Ga}}\left[\frac1{(1+z\int_Xa(x)d\bar\ga(x))^\th}\right],
\endalign
$$
and Theorem follows.
\enddemo         
    
\smallskip\noindent
{\bf Remarks. 1.} It follows from the known results on the Markov--Krein transform
that the distribution $\mu_a$ of a linear functional $f_a$ is absolutely continuous
(see~\cite{CR90} for an explicit formula for the density).  

{\bf 2.} See~\cite{Ts97} for a generalization of the Markov--Krein
identity for the distribution of linear functionals with respect to the two parameter
Dirichlet process, and~\cite{KTs98} for similar results on the common distributions of
several linear functionals of the Dirichlet process.
\smallskip

\subhead 7. The gamma measure as a weak limit of laws of $\al$-stable processes 
when $\alpha$ tends to zero
\endsubhead

For $\al>0$,
let $P^\al$ denote the law of the standard $\al$-stable process
on $(X,\nu)$, i.e.
a homogeneous process with L\'evy measure $s^{-\al-1}ds$, $s>0$. The Laplace transform
of this process equals
$$
\Bbb E^\al\left[\exp\left(-\int_Xa(x)d\eta(x)\right)\right]=
\exp\left(-\int_Xa(x)^\al d\nu(x)\right).
\tag 7.1
$$  
We introduce the measure $P^\al_k$ on $D$ which has the density with respect to $P^\al$,
$$
\frac{dP^\al_k}{dP^\al}(\eta)=\frac{\exp\left(-\frac k\beta\,\eta(X)\right)}
{\Bbb E^\al\left[\exp\left(-\frac k\beta\,\eta(X)\right)\right]},\qquad\text{where }
\beta=\al^{1/\al}.
$$
Consider a function $k(\al)$ such that  $\lim_{\al\to0}\frac{k(\al)^\al}{\Ga(1-\al)}=1$.
Then the measures $P^\al_{k(\al)}$ weakly converge to $P_\Ga$ when $\al\to0$ 
(see~\cite{GGV83, Li83, VY95}). 
This 
important result is a key point of the construction of two-parameter Poisson--Dirichlet 
distributions $PD(\al,\th)$ (\cite{PY97}). In particular, one obtains as a corollary that
for a fixed $\th\ne0$, the distributions $PD(\al,\th)$ converge to $PD(0,\th)=PD(\th)$ 
when $\al\to0$.

\Refs\widestnumber\key{GGV85} 

\ref \key CR90
\by D.M.Cifarelli, E.Regazzini
\paper Some remarks on the distribution functions of means
of a Dirichlet process
\jour Ann. Statist.
\vol 18
\yr 1990
\pages 429--442
\endref

\ref \key DK96
\by P.Diaconis, J.Kemperman
\paper Some New Tools for Dirichlet Priors
\inbook Bayesian Statistics 5
\yr 1996
\pages 97--106
\eds J.M.Ber\-nardo, J.O.Berger, A.P.Dawid and A.F.M.Smith
\publ Oxford University Press
\endref

\ref\key Fe73
\by T.S.Ferguson
\paper A Bayesian analysis of some nonparametric problems
\jour The Annals of Statistics
\vol 1
\yr 1973
\pages 209--230
\endref

\ref\key FK72
\by T.S.Ferguson, M.J.Klass
\paper A representation of independent increment processes without gaussian
components
\jour The Annals of Mathematical Statistics
\vol 43
\yr 1972
\pages 1634--1643
\endref

\ref\key GGV83
\by I.M.Gelfand, M.I.Graev, A.M.Vershik
\paper Commutative model of the representation of the group of currents
$SL(2,\Bbb R)^X$ connected with a unipotent subgroup
\jour Funct. Anal. 
\vol 17, 2
\yr 1983
\pages 80--82
\endref

\ref\key GGV85
\by I.M.Gelfand, M.I.Graev, A.M.Vershik
\paper Models of representations of current groups
\inbook Representations of Lie groups and Lie algebras  
\ed A.A.Kirillov
\publ Akad\'emiai Kiad\'o
\publaddr Budapest
\yr 1985
\pages 121--179
\endref

\ref\key Ke98
\by S.V.Kerov
\paper Interlacing Measures
\jour Amer. Math. Soc. Transl. (2)
\vol 181
\yr 1998
\pages 35--83
\endref

\ref\key KTs98
\by S.V.Kerov, N.V.Tsilevich
\paper The Markov--Krein correspondence in several dimensions
\jour PDMI Preprint 
\vol 1
\yr 1998
\endref

\ref\key Ki75
\by J.F.C.Kingman
\paper Random discrete distributions
\jour J.Roy.Statist.Soc. B
\vol 37
\yr 1975
\pages 1--22
\endref

\ref\key Ki93
\by J.F.C.Kingman
\book Poisson Processes
\publ Clarendon Press
\publaddr Oxford
\yr 1993
\endref

\ref\key Li83 
\by M.Lifschits
\paper Invariant measures generated by random fields with independent
values
\jour Funct. Anal.
\vol 19, 4
\yr 1983
\pages 92--92
\endref

\ref\key Lu??
\by Lukacs
\endref

\ref\key PY97
\by J.Pitman, M.Yor
\paper The two-parameter Poisson--Dirichlet distribution derived from a
stable subordinator
\jour Ann. Prob.
\yr 1997
\vol 25
\pages 855--900
\endref

\ref\key Ts97
\by N.V.Tsilevich
\paper Distributions of mean values for some random measures
\jour Zapiski nauchn. seminarov POMI
\vol 240
\yr 1997
\pages 268--279
\endref

\ref\key Ve77
\by A.M.Vershik
\paper Multivalued mappings with invariant measure (polymorphisms) and 
Mar\-kov operators 
\jour Zapiski nauchnyh seminarov LOMI
\vol 72
\yr 1977
\pages 26--61
\endref

\ref\key VY95
\by A.Vershik, M.Yor
\paper Multiplicativit\'e du processus gamma et \'etude asymptotique des
lois stables d'indice $\al$, lorsque $\al$ tend vers $0$
\jour Pr\'epublication du Laboratoire de probabilit\'es de l'Universit\'e
Paris VI
\vol 289
\yr 1995
\pages 1--10
\endref


\endRefs
\enddocument
