A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach
Christine Choirat, Christian Hess, and Raffaello Seri
Source: Ann. Probab. Volume 31, Number 1
(2003), 63-92.
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 $\sigma$-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.
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Keywords: Birkhoff ergodic theorem; stationary sequences; normal integrands; measurable set-valued maps; epigraphical convergence; set convergence; strong consistency of estimators
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1046294304
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Mathematical Reviews number (MathSciNet): MR1959786
Zentralblatt MATH identifier: 1015.60029
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JEUX, CONTRÔLE UNIVERSITÉ PARIS DAUPHINE
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