A counterexample concerning uniform ergodic theorems for a class of functions
It is proved that \textit{V. N. Vapnik}'s and \textit{A. Ya. Chervonenkis}' [Theory Probab. Appl. 16, 264-280 (1971); translation from Teor. Veroyatn. Primen. 16, 264-279 (1971; Zbl 0247.60005)] and \textit{M. Talagrand}'s [Ann. Probab. 15, 837-870 (1987; Zbl 0632.60024)] conditions do not imply convergence of relative frequencies to the probability \(P(C)\) for every set \(C \in {\mathcal L} \subset {\mathcal F}\) if one replaces the sequence of independent random variables with values in measurable space \(({\mathcal X}, {\mathcal F})\) by an ergodic stationary random process defined on \(({\mathcal X}, {\mathcal F})\) with marginal distribution \(P\).
scientific article; zbMATH DE number 796377