Central limit theorems for stochastic processes under random entropy conditions (Q1077074)

Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array \((\phi_{nj})\) of stochastic processes, indexed by a set S, to a sample- continuous Gaussian process, when the array satisfies a ''random entropy'' condition, analogous to one used by \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037)] for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-Červonenkis property and each \(\phi_{nj}\) is of the form \(\phi_{nj}(f)=\int fd\nu_{nj}\) for some reasonable random finite signed measure \(\nu_{nj}.\) As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik- Červonenkis classes. Special cases include \textit{Yu. V. Prokhorov}'s [Teor. Veroyatn. Primen. 1, 177-237 (1956; Zbl 0075.290); English translation in Theory Probab. Appl. 1, 188-214 (1956)] central limit theorem for processes on the unit interval, \textit{M. J. Wichura}'s [Ann. Math. Stat. 40, 681-687 (1969; Zbl 0214.177)] sufficient conditions for weak convergence of multiparameter processes, \textit{D. Pollard}'s [J. Aust. Math. Soc., Ser. A 33, 235-248 (1982; Zbl 0504.60023)] central limit theorem for empirical processes, and \textit{G. R. Shorack}'s [Stat. Neerl. 33, 169-189 (1979; Zbl 0436.60009)] theorems on weighted empirical processes.