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

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  • Xia Chen

    (Northwestern University)

Abstract

Let {X n } 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.

Suggested Citation

  • Xia Chen, 1999. "The Law of the Iterated Logarithm for Functionals of Harris Recurrent Markov Chains: Self Normalization," Journal of Theoretical Probability, Springer, vol. 12(2), pages 421-445, April.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:2:d:10.1023_a:1021630228280
    DOI: 10.1023/A:1021630228280
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    References listed on IDEAS

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    1. A. de Acosta & Xia Chen, 1998. "Moderate Deviations for Empirical Measures of Markov Chains: Upper Bounds," Journal of Theoretical Probability, Springer, vol. 11(4), pages 1075-1110, October.
    2. Chen, Xia, 1999. "Some dichotomy results for functionals of Harris recurrent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 211-236, September.
    3. Evarist Giné & David M. Mason, 1998. "On the LIL for Self-Normalized Sums of IID Random Variables," Journal of Theoretical Probability, Springer, vol. 11(2), pages 351-370, April.
    4. Csörgo, Miklós & Shao, Qi-Man, 1994. "A self-normalized Erdos--Rényi type strong law of large numbers," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 187-196, April.
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