The Annals of Probability

How Often Does a Harris Recurrent Markov Chain Recur?

Xia Chen

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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.

Article information

Source
Ann. Probab., Volume 27, Number 3 (1999), 1324-1346.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677449

Digital Object Identifier
doi:10.1214/aop/1022677449

Mathematical Reviews number (MathSciNet)
MR1733150

Zentralblatt MATH identifier
0981.60023

Subjects
Primary: 60F10: Large deviations 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Markov chain Harris recurrence regularity invariant measure additive functional

Citation

Chen, Xia. How Often Does a Harris Recurrent Markov Chain Recur?. Ann. Probab. 27 (1999), no. 3, 1324--1346. doi:10.1214/aop/1022677449. https://projecteuclid.org/euclid.aop/1022677449


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