LetSn 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(Sn/sn) converges almost surely to ?-88f(x)dF(x). We also obtain strong approximation forH(n)=?k=1nk-1f(Sk/sk)=logn ?-88f(x)dF(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 ?-mixing random variables.
Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.