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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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