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.
Publié le : 2003-01-14
Classification:
Birkhoff ergodic theorem,
stationary sequences,
normal integrands,
measurable set-valued maps,
epigraphical convergence,
set convergence,
strong consistency of estimators,
60F17,
28D05,
60G10,
37A30,
62F12,
49J35,
26E25,
28B20,
52A20
@article{1046294304,
author = {Choirat, Christine and Hess, Christian and Seri, Raffaello},
title = {A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 63-92},
language = {en},
url = {http://dml.mathdoc.fr/item/1046294304}
}
Choirat, Christine; Hess, Christian; Seri, Raffaello. A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Ann. Probab., Tome 31 (2003) no. 1, pp. 63-92. http://gdmltest.u-ga.fr/item/1046294304/