How often does a Harris recurrent Markov chain recur?

• 成果类型: Article
• 署名作者: Chen, X
• 署名单位: Northwestern University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677449
• 发表日期: 1999
• 页码: 1324-1346
• 关键词: iterated logarithm; random-walks; Levy processes; local-times; functionals; deviations; bounds; LAWS

摘要

Let {X-n}(n greater than or equal to 0) be a Harris recurrent Markov chain with state space (E, 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 (n)(k=1)Sigma f(X-/e), n=1, 2,... in the forms of both weak and strong convergences. In particular, the frequency of {X-n}(n greater than or equal to 0) visiting a given set A is an element of E with 0 < pi(A) < +infinity 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.