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

• Christine Choirat ^[1] ; Christian Hess ^[1] ; Raffaello Seri ^[2]
  1. [1] Paris Dauphine University

     Paris Dauphine University

     París, Francia

     
  2. [2] CREST-LFA
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 1, 2003, págs. 63-92
• Idioma: inglés
• DOI: 10.1214/aop/1046294304
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.