Central limit theorems for C(S)-valued random variables

AbstractLet C(S) be the space of real-valued continuous functions on a compact metric space S. Let {Xn, n ⩾ 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and suptϵs E[X12(t)] = 1. We show that the measures induced by (X1 + ··· + Xn) n−12 converge weakly to a Gaussian measure on C(S) under different conditions on X1, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors