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The Law of the Iterated Logarithm for Functionals of Harris Recurrent Markov Chains: Self Normalization

Authors:

Xia Chen

Institution:

(1) Department of Mathematics, Northwestern University, Evanston, Illinois, 60208

Abstract:

Let {X _n} _n0 be a Harris recurrent Markov chain with state space E, transition probability P(x, A) and invariant measure pgr

, and let f be a real measurable function on E. We prove that with probability one,

$\mathop {\lim \sup }\limits_{n \to \infty } \sum\limits_{k = 1}^n {f(X_k )/\sqrt {2\left( {\sum\limits_{k = 1}^n {f^2 (X_k )} } \right)\log \log \left( {\sum\limits_{k = 1}^n {f^2 (X_k )} } \right)} }$

$= \left( {1 + \left( {\int {f^2 (x)\pi (dx)} } \right)^{ - 1} \int {\sum\limits_{k = 1}^\infty {f(x)P^k f(x)\pi (dx)} } } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$

under some best possible conditions.

Keywords:

Law of the iterated logarithm Harris recurrent Markov chain invariant measure atom D-set

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