Convergence Rate in the Law of Logarithm for NA Random Variables

Let \X,X_n,n\geq1\ be a sequence of identically distributed NA random variables and set S_n=\sum_i=1^nX_i, n\geq 1. Let h(\cdot) be a positive nondecreasing function on (0,\infty) such that \int_1^\inftyth(t)^-1\md t=\infty. Denote Lt=\ln\max\e,t\, S_n=\sum_i=1^nX_i, \psi(t)=\int_1^tsh(s)^-1\md s, t\geq 1. In this paper, we prove that \sum_n=1^\inftynh(n)^-1\pr(\max_1\leq j\leq n|S_j|\geq (1+\varepsilon)\sigma\sqrt2nL\psi(n))<\infty, \forall\,\varepsilon>0 if and if \ep(X)=0 and \ep(X^2)=\sigma^2\in(0,\infty). The result partially extends and improves the theorems of 7.