A transient random walk on stochastic matrices with Dirichlet distributions
Suppose \(X_ 1, X_ 2, \ldots, X_ n, \ldots\) are stochastic \(d \times d\) matrices, which are random in the sense that \(X_ k\) has \(d\) independent rows and each row has a Dirichlet distribution. If \(X_ n\) \((n \geq 1\)) are independent and identically distributed, then it is shown that the product \(X_ n X_{n - 1} \cdots X_ 1\) converges almost surely to a random stochastic matrix with \(d\) identical rows which also have a Dirichlet distribution. There is a symmetry condition: if the \(d\) parameters of the Dirichlet distribution of the rows of \(X_ k\) are put in a \(d \times d\) matrix \(A\), then it is assumed that the sum of the entries on row \(j\) is the same as the sum of the entries on column \(j\). The parameters of the limiting Dirichlet distribution are then given by these \(d\) row (or column) sums. This generalizes a result of the reviewer [J. Appl. Probab. 23, 1019-1024 (1986; Zbl 0606.60012)] who treated the case \(d = 2\) for a matrix \(A\) with four identical entries.
scientific article; zbMATH DE number 567191