A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Choirat, C., Hess, C., & Seri, R. Ann. Probab., 31(1):63–92, Institute of Mathematical Statistics, January, 2003.
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Project Euclid - mathematics and statistics online
@ARTICLE{Choirat2003-hp,
  title     = "A functional version of the Birkhoff ergodic theorem for a
               normal integrand: A variational approach",
  author    = "Choirat, Christine and Hess, Christian and Seri, Raffaello",
  abstract  = "Project Euclid - mathematics and statistics online",
  journal   = "Ann. Probab.",
  publisher = "Institute of Mathematical Statistics",
  volume    =  31,
  number    =  1,
  pages     = "63--92",
  month     =  jan,
  year      =  2003,
  keywords  = "Birkhoff ergodic theorem; stationary sequences; normal
               integrands; measurable set-valued maps; epigraphical
               convergence; set convergence; strong consistency of estimators"
}

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