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Convergence time to equilibrium for large finite Markov chains. (Russian. English summary) Zbl 0969.60073
Let a sequence $$L(N)$$ of finite irreducible and aperiodic Markov chains with state space $$X(N)$$ be given. Let $$\pi(N)= (\pi_\alpha (N)$$, $$\alpha\in X(N))$$ and $$P_N$$ be the stationary distribution and matrix of transition probabilities of $$L(N)$$. For any probability distributions $$\nu$$ and $$\rho$$, put $$\|\nu- \rho\|= \sup_{B\subset X(N)}|\nu(B)- \rho(B) |$$. A function $$T(N)$$ is said to be the convergence time to equilibrium if $$\sup_\mu \|\mu P_N^{T(N) \psi(N)}-\pi (N)\|\to 0$$ as $$N\to\infty$$ and $$\psi(N) \to\infty$$. For sequences of truncations of some countable Markov chain $$L$$, the author finds the convergence time to equilibrium in terms of the Lyapunov function of $$L$$. The results are applied to queueing systems with limited number of customers.
##### MSC:
 60J05 Discrete-time Markov processes on general state spaces 37A50 Dynamical systems and their relations with probability theory and stochastic processes
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