# Search Results

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

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

Let*S*
_{n} be the partial sums of ?-mixing stationary random variables and let*f(x)* be a real function. In this note we give sufficient conditions under which the logarithmic average of*f(S*
_{n}/s_{n}) converges almost surely to ?_{-8}
^{8}
*f(x)d*F*(x)*. We also obtain strong approximation for*H(n)*=?_{k=1}
^{n}
*k*
^{-1}
*f(S*
_{k}/s*k*)=log*n* ?_{-8}
^{8}
*f(x)d*F*(x)* which will imply the asymptotic normality of*H(n)*/log^{1/2}
*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.

Summary In this note we prove an almost sure limit theorem for the products of *U*-statistics.

## Abstract

*X*:

_{n}*n*≧ 1} be a sequence of dependent random variables and let {

*w*: 1 ≦

_{nk}*k*≦

*n, n*≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

## Abstract

Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of Csáki, Földes and Révész is revisited and slightly improved.

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.

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