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On the Central Limit Theorem for Geometrically Ergodic Markov Chains

Journal article
Authors Olle Häggström
Published in Probability theory and related fields
Volume 132
Issue 1
Pages 74-82
ISSN 0178-8051
Publication year 2005
Published at Department of Mathematical Sciences, Mathematical Statistics
Pages 74-82
Language en
Subject categories Mathematics


Let X-0,X-1,... be a geometrically ergodic Markov chain with state space X and stationary distribution pi. It is known that if h:X -> R satisfies pi(vertical bar h vertical bar(2+epsilon)) < infinity for some epsilon > 0, then the normalized sums of the X-i's obey a central limit theorem. Here we show, by means of a counterexample, that the condition pi(vertical bar h vertical bar(2+epsilon)) < infinity cannot be weakened to only assuming a finite second moment, i.e., pi(h(2)) < infinity.

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