Convergence in distribution of products of \(d\times{}d\) random matrices
Let \(A_ 1,A_ 2,\ldots\) be an i.i.d. sequence of \(d\times d\) real matrices. The authors study the random walk \(M_ n=A_ 1A_ 2\ldots A_ n\), \(n=1,2,\ldots,\) and its limiting behavior. In particular, the paper aims at giving easily verifiable conditions on the (support \(S_ \mu\) of the) distribution \(\mu\) of the \(A_ i\)'s for the random walk to converge in distribution. Let \(S\) be the semigroup of matrices generated by \(S_ \mu\); \(S\) is the natural state space of the Markov chain \(\{M_ n\}\). The methods used are based on results about convolution sequences of probability measures on the topological semigroup \(S\), cf. [\textit{A. Mukherjea}, Trans. Am. Math. Soc. 303, 395-411 (1987; Zbl 0638.60007) and Probabilistic analysis and related topics, Vol. 2, 143-203 (1979; Zbl 0453.60015)]. The paper extends certain previous results on bistochastic matrices (i.e., matrices with non-negative elements whose row sums and column sums are all equal to 1) by \textit{M. Rosenblatt} [J. Math. Anal. Appl. 11, 1-10 (1965; Zbl 0203.195)]. For non-negative matrices the authors obtain, e.g., the following result: Let \(J\) be the set of all matrices in \(S\) which have at least one zero row or one zero column or both, and let \(m(S)\) be the set of matrices in \(S\) with the minimal rank. Suppose that the intersection of \(m(S)\) with the complement \(J^ c\) of \(J\) is non-empty. Then the sequence \(\{\mu^ n\}\) is tight (implying, e.g., that it converges weakly in the Cesáro sense) if and only if \(S\) is compact. --- Furthermore, if instead of assuming \(m(S)\cap J^ c\neq\emptyset\) the assumption is only that \(0\notin S\), then the tightness of \(\{\mu^ n\}\) implies that \(S\) is nearly compact, that is, there is \(L>0\) such that for each \(x\in S\), \(\min_{i,j}x_{ij}\leq L\). Conversely, if \(\mu(J)=0\) and for some positive integer \(N\), \[ \mu^ N\{\text{the strictly positive matrices in } S\}>0, \] then the near compactness of \(S\) implies the tightness of \(\{\mu^ n\}\). For bistochastic \(d\times d\) matrices the authors make a detailed study of the cases \(d=2,3,4.\) Necessary and sufficient conditions for convergence are also found for a general \(d\). The proofs exploit efficiently the algebraic structure of the compact topological semigroup \(B_ d\) of \(d\times d\) bistochastic matrices, the salient fact being that any closed subsemigroup of \(B_ d\) admits a completely simple kernel which is a finite group.
convolution sequences of probability measures
scientific article; zbMATH DE number 61582
scientific article