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

Abstract

Let {Xn} n≥0 be a Harris recurrent Markov chain with state space E, transition probability P(x, A) and invariant measure π, 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.