A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach

• 成果类型: Article
• 署名作者: Choirat, C; Hess, C; Seri, R
• 署名单位: Universite PSL; Universite Paris-Dauphine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 63-92
• 关键词: large numbers; Conditional expectations; epi-convergence; uniform law; Consistency

摘要

In this paper, we prove a new version of the Birkhoff ergodic theorem (BET) for random variables depending on a parameter (alias integrands). This involves variational convergences, namely epigraphical, hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands. We also have to establish the measurability of the epigraphical lower and upper limits with respect to the or-field of invariant subsets. From the main result, applications to uniform versions of the BET to sequences of random sets and to the strong consistency of estimators are briefly derived.