The functional central limit theorem for strongly mixing processes

Abstract

Let (XJi E 7l be a strictly stationary and strongly mixing sequence of Rd-valued zero-mean random variables. Let be the sequence of mixing coefficients. We define the strong mixing function a by and we denote by Q the quantile function of Xo ~, which is the inverse function of t P ( X0| > t). The main result of this paper is that the functional central limit theorem holds whenever the following condition is fulfilled: where f ’ -1 denotes the inverse of the monotonic function f. Note that this condition is equivalent to the usual condition E (X~) oo for m-dependent sequences. Moreover, for any a > 1, we construct a sequence with strong mixing coefficients an of the order of n-a such that the CLT does not hold as soon as condition (* ) is violated.