Choirat, Christine; Hess, Christian; Seri, Raffaello A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. (English) Zbl 1015.60029 Ann. Probab. 31, No. 1, 63-92 (2003). Summary: 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. Cited in 20 Documents MSC: 60F17 Functional limit theorems; invariance principles 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 28D05 Measure-preserving transformations 37A99 Ergodic theory 60G10 Stationary stochastic processes 62F12 Asymptotic properties of parametric estimators Keywords:stationary sequences; normal integrands; set convergence; Birkhoff ergodic theorem × Cite Format Result Cite Review PDF Full Text: DOI References: [1] ALIPRANTIS, C. D. and BORDER, K. C. (1999). Infinite Dimensional Analy sis, 2nd ed. Springer, Berlin. [2] AMEMIy A, T. (1985). Advanced Econometric Methods. Harvard Univ. Press. [3] ANDREWS, D. W. K. (1987). Consistency in nonlinear econometric models: a generic uniform law of large numbers. Econometrica 55 1465-1471. JSTOR: · Zbl 0646.62101 · doi:10.2307/1913568 [4] ARTSTEIN, Z. and HART, S. (1981). Law of large numbers for random sets and allocation processes. Math. Oper. Res. 6 485-492. 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