Almost Sure Central Limit Theorem for Strictly Stationary Processes

On any aperiodic measure preserving system, there exists a square integrable function such that the associated stationary process satifies the Almost Sure Central Limit Theorem. Introduction The Almost Sure Central Limit Theorem (ASCLT), first formulated by Lévy in [9], has been studied by various authors at the end of the eighties ([6], [3], [10], [8]). This theorem gives conditions under which, for a sequence of random variables satisfying the Central Limit Theorem (CLT), the Gaussian asymptotic behaviour can be observed along individual trajectory of the process. In the Lacey and Philipp note [8], the ASCLT is stated under optimal hypotheses, and the proof is short and clear. Here is their result. (If x is a real number, notation δ(x) will be used for the Dirac mass at point x.) Theorem. Let (Xn)n≥1 be an independent and identically distributed sequence of square integrable real random variables with E(Xn) = 0 and E(X n) = 1. Almost surely, the sequence of probability distributions ( 1 logn n ∑ k=1 1 k δ ( X1 +X2 + · · ·+Xk √ k )) n≥1 converges weakly to the Gaussian law N(0, 1). Several authors, including Berkes and Dehling ([2]), Atlagh and Weber ([1]), and Lacey have observed that for i.i.d. sequences, the finite second moment condition is necessary for the ASCLT. So, in this context, necessary and sufficient conditions for the CLT and the ASCLT are the same. This paper is a contribution to the study of the general case of strictly stationary sequences. The question of the CLT for strictly stationary processes has been extensively studied, from various points of view. Given a probability measure preserving dynamical system (Ω, T , μ, T ) and a real measurable function f on Ω we say that Received by the editors June 14, 1998 and, in revised form, July 22, 1998. 1991 Mathematics Subject Classification. Primary 28D05, 60G10, 60F05.