LIMIT THEOREMS FOR ADDITIVE FUNCTIONALS OF A MARKOV CHAIN

• 成果类型: Article
• 署名作者: Jara, Milton; Komorowski, Tomasz; Olla, Stefano
• 署名单位: Universite PSL; Universite Paris-Dauphine; Maria Curie-Sklodowska University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP610
• 发表日期: 2009
• 页码: 2270-2300
• 关键词: random schrodinger-equation

摘要

Consider a Markov chain {X(n)}(n >= 0) with an ergodic probability measure pi. Let be a function on the state space of the chain, with a-tails with respect to pi, alpha is an element of (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N(1/alpha) Sigma(N)(n) Psi(X(n)) to an a-stable law. A martingale approximation approach and a coupling approach give two different sets of conditions. We extend these results to continuous time Markov jump processes X(t), whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N(-1/alpha) integral(Nt)(0) V (X(s))ds to a stable process. The result is applied to show that an appropriately scaled limit of solutions of a linear Boltzman equation is a solution of the fractional diffusion equation.