Central limit theorems for stochastic processes under random entropy conditions

Summary Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array ( _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 Giné and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-ervonenkis property and each _nj (f)fdnj is of the form njc for some reasonable random finite signed measure v _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-ervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.Research supported by an NSF Postdoctoral Fellowship, grant no. MCS83-111686