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

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 $n\to\infty$.