A functional central limit theorem for strongly mixing sequences of random variables

Abstract

We prove a functional central limit theorem for a class of strongly mixing sequences of random variables. Stationarity is not assumed, but the variances of the partial sums must grow linearly. Our theorem extends previous results by supplying sufficient conditions for the weak convergence of the partial sum process to the Wiener process under less restrictive moment assumptions. For instance, if {Mathematical expression} for some ε>0, and the mixing rate is exponential, then this functional c.l.t. holds. Under the weaker assumption with ε=0, the c.l.t. may fail to hold, and it is possible that the c.l.t. is satisfied, but the sequence of partial sum processes is not tight. © 1985 Springer-Verlag.