Strassen type limit points for moving averages of a wiener process

Let { W(s) ; 8 ≥ 0} be a standard Wiener process, and let β N = (2a N ( log ( N/a N ) + log log N ) ‐1/2 , 0 < α N ≤ N < ∞, where α N ↑ and α N / N is a non‐increasing function of N , and define γ N ( t ) = β N [ W ( n N + ta N ) − W ( n N )), 0 ≥ t ≥ 1, with n N = N – a N . Let K = { x ϵ C [0,1]: x is absolutely continuous, x (0) = 0 and\documentclass{article}\pagestyle{empty}\begin{document}$$ \[\int_0^1 {\left({\frac{{dx}} {{dt}}} \right)^2 dt \le 1} \] $$\end{document} }. We prove that, with probability one, the sequence of functions {γ N ( t ), t ϵ [0,1]} is relatively compact in C [0,1] with respect to the sup norm ||·||, and its set of limit points is K . With a N = N , our result reduces to Strassen's well‐known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.