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Maximum asymptotic variance of sums of finite Markov chains. (English) Zbl 0991.60061

The present note considers a reversible Markov chain \((X_n)\) defined on the finite state space \(E\) with irreducible and aperiodic transition matrix \(P=(p_{ij})\) and stationary distribution \(\pi\). Let \(f:E\to R\) satisfy \(\min f(E)=0\), \(\max f(E)=1\), and \(\mu=\int fd\pi\). The purpose of this note is to study the asymptotic variance of the empirical mean sequence \((f_n)={1\over n} \sum^n_{k=1} f(X_k)\), that is known to converge a.s. to \(\mu\). The main theorem of the article constructs an optimal bound for the asymptotic variance of the empirical mean sequence, and proves that this bound depends only on \(\mu\), on the second largest eigenvalue \(\lambda\) of the chain \((X_n)\), and on the endpoints of the support of \(f(E)\).

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C40 Numerical analysis or methods applied to Markov chains
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References:

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