ON CONVERGENCE-RATES OF AVERAGES OF WEAKLY DEPENDENT RANDOM-VARIABLES

Let {X(n)}n=1infinity be a sequence of random variables on a probability space, r > 1 and the delayed sum S(m,n) = SIGMA(k=m+1)(m+n)X(k), where m greater-than-or-equal-to 0 and n greater-than-or-equal-to 1. Further, let the function rho(n) = sup(m greater-than-or-equal-to 0\\(1/n)S(m,n)\\r(r) satisfy SIGMA(n=1)(infinity)rho(n)/n < infinity, where \\X\\(r) denotes the rth mean of random variable X. Then we prove that for 2k less-than-or-equal-to n < 2k+1 and for m greater-than-or-equal-to 0, \\(1/n)S(m,n)\\r(r) less-than-or-equal-to c(r){rho(2k) + SIGMA(j=1)k1/2((1 + c(r))/2r)j-1rho(2k-j)} = upsilon(k) = o(k-1) as k --> infinity, where c(r) = 2r-1. We also prove that for m greater-than-or-equal-to 0, p greater-than-or-equal-to 1, and epsilon > 0, P{sup(n greater-than-or-equal-to 2p)((1/n)\S(m,n)\) > epsilon} less-than-or-equal-to (2/epsilon(r))SIGMA(k=p)(infinity)upsilon(k) --> 0 as p --> infinity).