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Weak convergence of matrices of transition probabilities for the conditioned Markov chains

Z. I. Bežaeva

Moscow

Abstract: Let $\zeta_t=(\xi_t,\eta_t)$ be a two-dimensional countable Markov chain. The component $\xi_t\,(t=1\div n)$ may be considered as a conditioned Markov chain with respect to the conditional probability measure $\mathbf P\{\cdot\mid\eta_1,\dots,\eta_n\}$. We prove that under some assumptions all components of the matrix of transition probabilities of conditioned Markov chain converge weakly to the corresponding limits when $n\to\infty$.

Received: 10.01.1980

 English version: Theory of Probability and its Applications, 1982, 27:1, 59–68

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