MATHEMATICS

V. N. TUTUBALIN

LIMITING BEHAVIOR OF COMPOSITIONS OF MEASURES IN SOME SYMMETRIC SPACES

(Presented by Academician A. N. Kolmogorov on November 3, 1961)

1. Consider a symmetric space \(M\) on which a group of motions \(G\) acts. Denote by \(K\) the stationary subgroup of some point \(e \in M\). We shall call a probability measure \(\mu\), defined on all Borel subsets of \(M\), symmetric if for any Borel set \(\Gamma \subset M\) and any \(k \in K\) the equality \(\mu(\Gamma)=\mu(k\Gamma)\) holds. With a sequence of symmetric measures \(\mu_1,\mu_2,\ldots,\mu_m,\ldots\) there is associated a Markov chain \(x(m)\), \(m=0,1,\ldots\), invariant with respect to motions, for which the probability \(P_{m+1}^m(x,\Gamma)\) of passing during the time \((m,m+1)\) from the point \(x\) into the set \(\Gamma\) is equal to \(\mu_{m+1}(g_x^{-1}\Gamma)\), where \(g_x\) is any motion carrying \(e\) into \(x\). The transition probabilities for the time \((0,m)\) are determined by the recurrent formula:

\[ P_m^0(x,\Gamma)=\int_M P_{m-1}^0(x,dy)\,P_m^{m-1}(y,\Gamma). \]

We shall call the measure \(P_m^0(e,\Gamma)\) the composition \(\mu_1 * \ldots * \mu_m\) of the measures \(\mu_i\), \(i=1,\ldots,m\). In the case when \(\mu_1=\ldots=\mu_m=\mu\), we shall write \(\mu_1 * \ldots * \mu_m=\mu^m\). We shall consider the question of the behavior of the measure \(\mu^m\) as \(m\to\infty\). The results are formulated for the case when \(G\) is the complex unimodular group. Analogous results are valid for other classical complex groups.

2. Let \(M\) be the set of positive-definite Hermitian unimodular matrices of order \(n\); \(G\) the group of unimodular matrices acting on \(M\) according to the formula \(g:x\to gxg^{-}\) \((x\in M)\). The stationary subgroup of the identity matrix \(e\) is the unitary group \(K\).

Denote by \(D\) the set of points \(t=(t_1,\ldots,t_n)\) of Euclidean space defined by the equality
\[ D=\{t:t_1+\ldots+t_n=0,\ t_1\ge t_2\ge\ldots\ge t_n\}. \]
To each matrix \(x\in M\) we associate the collection \(t(x)\) of logarithms of its eigenvalues, arranged in decreasing order. Obviously, \(t(x)\in D\) for any \(x\in M\). Every measure \(\mu\) on \(M\) induces a measure \(\bar\mu\) on \(D\) by the formula \(\bar\mu(A)=\mu\{x:t(x)\in A\}\). Moreover, an arbitrary measure on \(D\) is induced by one and only one symmetric measure on \(M\). In what follows we shall denote a symmetric measure on \(M\) and the measure induced by it on \(D\) by the same letter.

3. Let \(\Phi(\rho,x)\) be a bounded zonal function on \(M\) (\(\rho=(\rho_1,\ldots,\rho_n)\) is a complex vector), i.e., an eigenfunction of the Laplace operators on \(M\), invariant with respect to the stationary subgroup \(K\) \((^1)\). The function \(\Phi(\rho,x)\) is uniquely determined by the set of eigenvalues (which are expressed in terms of \(\rho\)) and by the condition \(\Phi(\rho,e)=1\). We shall call the function

\[ f_\mu(\rho)=\int_M \Phi(\rho,x)\,\mu(dx)=\int_D \Phi(\rho,t)\,\mu(dt), \]

where \(\Phi(\rho,t)\) is the value of the function \(\Phi(\rho,x)\) on matrices \(x\in M\) such that \(t(x)=t\), the characteristic function of the measure \(\mu\). With the invariant Markov chain \(x(m)\) (see item 1) there are associated the operators \(T_m^0\), defined by the formula
\[ T_m^0\varphi(x)=\int_M \varphi(y)\,P_m^0(x,dy), \]
permutable-

... with all shifts. It can be shown that

\[ T_m^0\Phi(\rho,x)=f_{\mu_1}(\rho)\ldots f_{\mu_m}(\rho)\Phi(\rho,x). \]

Putting \(x=e\), it follows that

\[ f_{\mu_1*\ldots *\mu_m}(\rho)=\prod_{i=1}^{m} f_{\mu_i}(\rho). \]

For the case of the unimodular group \(G\) considered by us, the functions \(\Phi(\rho,t)\) have the form (see (2)):

\[ \Phi(\rho,t)= \frac{c\det\|e^{\rho_i t_j}\|} {\displaystyle \prod_{i<j}(\rho_i-\rho_j)\operatorname{sh}\frac{t_i-t_j}{2}}, \qquad i,j=1,\ldots,n. \]

4. Lemma. Let \(\rho=\sigma+i\tau\), where

\[ \sigma=\left\{\frac{n-1}{2},\frac{n-3}{2},\ldots,-\frac{n-1}{2}\right\}, \]

and \(\tau\) is a real vector. The functions \(\Phi(\rho,t)=\Phi(\sigma+i\tau,t)\) satisfy the following conditions:

\(1^\circ\). \(\Phi(\sigma+i\tau,t)\big|_{\tau=0}=1\) for any \(t\in D\); for any \(\tau\), \(\Phi(\sigma+i\tau,t)\) is bounded as a function of \(t\in D\).

\(2^\circ\). For any \(t\), \(\Phi(\sigma+i\tau,t)\) is a positive-definite function of \(\tau\).

\(3^\circ\). The inverse Fourier transform (in \(\tau\)) \(\widetilde{\Phi}_t\) of the function \(\Phi(\sigma+i\tau,t)\) (which, by \(1^\circ\) and \(2^\circ\), is a probability measure in Euclidean space) satisfies the condition: for any \(\varepsilon>0\),

\[ \lim_{m\to\infty}\mu^m\left\{t:\lim_{a\to\infty}\widetilde{\Phi}_t\{y:\|y-t\|>a\}>\varepsilon\right\}=0 \]

for any symmetric measure \(\mu\) on \(M\) (here \(\|y\|=\max_{1\le i\le n}|y_i|\)).

\(4^\circ\). For no fixed \(t\ne0\) does the equality

\[ \Phi(\sigma+i\tau,t)=e^{i\tau_k a_k}\varphi(\tau_1,\ldots,\tau_{k-1},\tau_{k+1},\ldots,\tau_n) \]

hold identically in \(\tau\), where \(\varphi\) does not depend on \(\tau_k\).

From conditions \(1^\circ\) and \(2^\circ\) it follows that, for any measure \(\mu\) on \(M\), the function (of \(\tau\) \(f_\mu(\sigma+i\tau)\) is the characteristic function (in the usual sense) of some probability measure \(\widetilde{\mu}\) in Euclidean space. Moreover,

\[ \widetilde{\mu^m}=(\widetilde{\mu})^m \]

(where the composition on the right is understood in the usual sense).

Denote by \(\widehat{\Gamma}_c\), where \(c=(c_1,\ldots,c_n)\), \(-\infty<c_i<\infty\), the set \(\{t:t_i<c_i,\ i=1,\ldots,n\}\) in Euclidean space, and by \(\Gamma_c\) the set \(\widehat{\Gamma}_c\cap D\).

Theorem. As \(m\to\infty\),

\[ \sup_c\left|\mu^m(\Gamma_c)-\widetilde{\mu}^{\,m}(\widehat{\Gamma}_c)\right|\to0. \]

Proof. There is the formula

\[ \widetilde{\mu}^{\,m}(\widehat{\Gamma}_c) = \int_D \widetilde{\Phi}_t(\widehat{\Gamma}_c)\,\mu^m(dt), \]

from which it follows that, under conditions \(1^\circ\)—\(3^\circ\) of the lemma, for any \(\varepsilon>0\) and \(a>a(\varepsilon)\),

\[ \widetilde{\mu}^{\,m}(\widehat{\Gamma}_{c-a}) \le \mu^m(\Gamma_c)+\varepsilon+\delta'_m, \]

\[ \widetilde{\mu}^{\,m}(\widehat{\Gamma}_{c+a}) \ge \mu^m(\Gamma_c)-\varepsilon+\delta''_m, \]

where \(\delta'_m\) and \(\delta''_m\) tend to zero as \(m\to\infty\), uniformly in \(c\) \((c\pm a=\{c_1\pm a,\ldots,c\pm a\})\). From condition \(4^\circ\) it follows that the measure \(\widetilde{\mu}\) does not concent...

is not concentrated on any plane of the form \(t_k=a_k\). Hence it follows that, for any \(a\),

\[ \lim_{m\to\infty}\sup_c \tilde{\mu}^{\,m}\{t:\ c_k-a<t_k<c_k+a\}=0. \]

Consequently,

\[ \lim_{m\to\infty}\sup_c \left|\tilde{\mu}^{\,m}(\tilde{\Gamma}_c)-\tilde{\mu}^{\,m}(\Gamma_{c\pm a})\right|=0. \]

The theorem is proved.

Remark. It is not hard to verify that for any measure \(\mu\) the measure \(\tilde{\mu}\) is concentrated on the plane \(t_1+\cdots+t_n=0\).

We shall say that the compositions of the measures \(\mu_1\) and \(\mu_2\) approach each other if

\[ \lim_{m\to\infty}\sup_c \left|\mu_1^m(\Gamma_c)-\mu_2^m(\Gamma_c)\right|=0. \]

Corollary 1. In order that the compositions of the measures \(\mu_1\) and \(\mu_2\) approach each other, it is necessary and sufficient that the compositions (in the ordinary sense) of the measures \(\tilde{\mu}_1\) and \(\tilde{\mu}_2\) approach each other, i.e., that

\[ \lim_{m\to\infty}\sup_c \left|\tilde{\mu}_1^{\,m}(\tilde{\Gamma}_c)-\tilde{\mu}_2^{\,m}(\tilde{\Gamma}_c)\right|=0. \]

Corollary 2. If the function \(f_\mu(\sigma+i\tau)\) has continuous second partial derivatives with respect to \(\tau_i,\tau_k\), then the probability distribution \(\mu^m\) of the vector \((t_1(x),\ldots,t_n(x))\) is asymptotically normal with parameters

\[ \left\{\frac{1}{i}\frac{\partial f_\mu}{\partial \tau_k}\bigg|_{\tau=0},\ k=1,\ldots,n\right\}, \qquad \left\{\left[-\frac{\partial^2 f_\mu}{\partial \tau_k\partial \tau_l} +\frac{\partial f_\mu}{\partial \tau_k}\frac{\partial f_\mu}{\partial \tau_l}\right]_{\tau=0},\ k,l=1,\ldots,n\right\}. \]

In conclusion the author expresses gratitude to E. B. Dynkin, under whose guidance the present work was carried out.

Moscow State University
named after M. V. Lomonosov

Received
18 X 1961

REFERENCES

1. F. A. Berezin, Trudy Moskov. Mat. Obshch., 6, 371 (1957).
2. I. M. Gel'fand, M. A. Naimark, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 36 (1950).