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July 1999 How Often Does a Harris Recurrent Markov Chain Recur?
Xia Chen
Ann. Probab. 27(3): 1324-1346 (July 1999). DOI: 10.1214/aop/1022677449

Abstract

Let ${X _n}_ {n\geq 0}$ be a Harris recurrent Markov chain with state space $(E,\mathscr{E})$, transition probability $P(x, A)$ and invariant measure $\pi$ . Given a nonnegative $\pi$-integrable function $f$ on $E$, the exact asymptotic order is given for the additive functionals

\sum_{k=1}^{n} f(X_k) \quad n=1,2,\dots

in the forms of both weak and strong convergences. In particular, the frequency of ${X_n}_{n \geq 0}$ visiting a given set $A\in \mathscr{E}$ with $0 < \pi (A) < + \infty$ is determined by taking $f = I_A$. Under the regularity assumption, the limits in our theorems are identified. The one- and two-dimensional random walks are taken as the examples of applications.

Citation

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Xia Chen. "How Often Does a Harris Recurrent Markov Chain Recur?." Ann. Probab. 27 (3) 1324 - 1346, July 1999. https://doi.org/10.1214/aop/1022677449

Information

Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0981.60023
MathSciNet: MR1733150
Digital Object Identifier: 10.1214/aop/1022677449

Subjects:
Primary: 60F10, 60J10

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 3 • July 1999
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