The Annals of Probability

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

Christine Choirat, Christian Hess, and Raffaello Seri

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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.

Article information

Source
Ann. Probab., Volume 31, Number 1 (2003), 63-92.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1046294304

Digital Object Identifier
doi:10.1214/aop/1046294304

Mathematical Reviews number (MathSciNet)
MR1959786

Zentralblatt MATH identifier
1015.60029

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 28D05: Measure-preserving transformations 60G10: Stationary processes 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 62F12: Asymptotic properties of estimators 49J35: Minimax problems 26E25: Set-valued functions [See also 28B20, 49J53, 54C60] {For nonsmooth analysis, see 49J52, 58Cxx, 90Cxx} 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]

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

Citation

Choirat, Christine; Hess, Christian; Seri, Raffaello. A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Ann. Probab. 31 (2003), no. 1, 63--92. doi:10.1214/aop/1046294304. https://projecteuclid.org/euclid.aop/1046294304


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  • 75775 PARIS, CEDEX 16 FRANCE E-MAIL: choirat@viab.ufrmd.dauphine.fr Christian.Hess@dauphine.fr R. SERI CREST-LFA TIMBRE J320 15 BD GABRIEL PÉRI 92245 MALAKOFF CEDEX FRANCE E.MAIL: seri@ensae.fr