MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES

Let {X(n); n >= 1} be a strictly stationary sequeuce of negatively associated random variables with mean zero and finite variance. Set S(n) = Sigma(n)(k=1) M(n) = max(k <= n) vertical bar S(k)vertical bar, n >= 1. Suppose sigma(2) - EX(1)(2) + 2 Sigma(infinity)(k=2) EX(1)X(k) (.0 < sigma < infinity)). We prove that for airy b > -1/2 if E vertical bar X vertical bar(2+delta) (0 < delta <= 1), then lim/epsilon SE arrow 0 epsilon(2b+1) Sigma(infinity)(n=1) (log log n)(b-1/2)/n(3/2) log n E {M(n) -sigma epsilon root 2n log log n}(+) + 2(-1/2-b)sigma E vertical bar N vertical bar(2(b+1))/(b + 1)(2b + 1) Sigma(infinity)(k=0) (-1)(k)/(2k + 1)(2(b+1)) and for any b > - 1/2, lim/epsilon NE arrow 0 epsilon(-2b+1) Sigma(infinity)(n=1) (log log n)(b)/n(3/2) log n E {sigma epsilon root pi(2)n/8 log log n - M(n)}+ = Gamma(b + 1/2)/root 2(b + 1) Sigma(infinity)(n=1) (-1)(k/)(2k + 1)(2b+2), Where Gamma(.) is the Gamma function and N stands for the standard normal random variable