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January 2003 A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach
Christine Choirat, Christian Hess, Raffaello Seri
Ann. Probab. 31(1): 63-92 (January 2003). DOI: 10.1214/aop/1046294304

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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Christine Choirat. Christian Hess. Raffaello Seri. "A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach." Ann. Probab. 31 (1) 63 - 92, January 2003. https://doi.org/10.1214/aop/1046294304

Information

Published: January 2003
First available in Project Euclid: 26 February 2003

zbMATH: 1015.60029
MathSciNet: MR1959786
Digital Object Identifier: 10.1214/aop/1046294304

Subjects:
Primary: 60F17
Secondary: 26E25 , 28B20 , 28D05 , 37A30 , 49J35 , 52A20 , 60G10 , 62F12

Keywords: Birkhoff ergodic theorem , epigraphical convergence , measurable set-valued maps , normal integrands , set convergence , Stationary sequences , strong consistency of estimators

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 1 • January 2003
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