Estimation of the Asymptotic Variance in the CLT for Markov Chains

Didier Chauveau (1) , Jean Diebolt (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

This paper is devoted to estimating the asymptotic variance in the central limit theorem (CLT) for Markov chains. We assume that the functional CLT for Markov chains applies for properly normalized partial-sum processes of functions of the chain, and study a continuous-time empirical variance process based on i.i.d. parallel chains. The centered empirical variance process, properly normalized, converges in distribution to a centered Gaussian process with known covariance function. This allows us to estimate the limiting variance and to control the fluctuations of the variance estimator after n steps. An application to Monte Carlo Markov chain (MCMC) convergence control is suggested.

Mots clés: Markov chains; Functional CLT; Asymptotic variance; Stochastic process; Itô process

Domaines: Statistiques [math.ST]

Didier Chauveau, Jean Diebolt. Estimation of the Asymptotic Variance in the CLT for Markov Chains. Stochastic Models, 2003, 19 (4), pp.449-465. ⟨10.1081/STM-120025399⟩. ⟨hal-00202574⟩