\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}). 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