Abstract
A sufficient condition ensuring weak ergodicity asr→∞ of productsP m,r ={p (m,r) ij }=P m+1 P m+2 ⋯P 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 (m,r)is /p (m,r)js →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.
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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.
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Seneta, E. Ergodicity for products of infinite stochastic matrices. J Theor Probab 6, 345–352 (1993). https://doi.org/10.1007/BF01047578
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DOI: https://doi.org/10.1007/BF01047578