A strong approximation for logarithmic averages of partial sums of random variables

Abstract

LetSn be the partial sums of ϱ-mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(Sn/σn) converges almost surely to ∫−∞∞f(x)dΦ(x). We also obtain strong approximation forH(n)=∑k=1nk−1f(Sk/σk)=logn ∫−∞∞f(x)dΦ(x) which will imply the asymptotic normality ofH(n)/log1/2n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for ϱ-mixing random variables.