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Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.

We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments of martingales. Several statistical applications to autoregressive and branching processes are also provided.

## 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

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.

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.