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

LetS_n be the partial sums of -mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S_n/_n) converges almost surely to _–f(x)d(x). We also obtain strong approximation forH(n)=_k=1ⁿk^–1f(S_k/k)=logn _–f(x)d(x) which will imply the asymptotic normality ofH(n)/log^1/2n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for -mixing random variables.