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BERKES, I. and HORVÁTH, L., almost sure invariance principles for logarithmic averages, Studia Sci. Math. Hungar. 33 (1997), 1-24. MR 98f :60054 Almost sure invariance principles for

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logarithmic averages, Math. Proc. Cambridge Philos. Soc. 112 (1992), 195-205. MR 93e :60057 Invariance principles for logarithmic averages Math. Proc. Cambridge Philos. Soc

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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/sn) converges almost surely to ?-8 8 f(x)dF(x). We also obtain strong approximation forH(n)=?k=1 n k -1 f(S k/sk)=logn ?-8 8 f(x)dF(x) which will imply the asymptotic normality ofH(n)/log1/2 n. 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.

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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.

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logarithmic averages, Studia Sci. Math. Hungar . 31 (1996), 187-196. MR 97b :60051 A strong approximation for logarithmic averages Studia Sci. Math. Hungar

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