Ergodicity for products of infinite stochastic matrices

A sufficient condition ensuring weak ergodicity asr of productsP_m,r={p_ij^(m,r)}=P_m+1P_m+2P_m+r formed from a sequence {P_k} of infinite stochastic matrices each of which contains no zero column, is given. The condition framed in terms of a generalization of Birkhoff's coefficient of ergodicity to such matrices, ensures also thatp_is^(m,r)/p_js^(m,r)1 asr uniformlyiss, for fixedi, j, m. The result, which relies partly on work of Gibert and Mukherjea,⁽⁴⁾ also generalizes a classical result of Kolmogorov.⁽⁶⁾ A corresponding discussion is given for backwards products.Forms part of results announced at the conference 50 years after Doeblin: Developments in the theory of Markov chains, Markov processes and sums of random variables held at Blaubeuren, Germany, November 2–7, 1991.