A Functional Version of the Birkhoff Ergodic Theorem for a Normal Integrand: A Variational Approach

Christine Choirat, Christian Hess, Raffaello Seri

January 2003

Abstract

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 $σ$ -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.

Type

Journal article

Publication

Annals of Probability, 31(1), 63-92