Iterated logarithm type behavior for weighted sums of i.i.d. random variables

摘要

For a sequence of i.i.d. mean 0 random variables {X, X.; n >= 1) with weighted partial SUMS S-n(X, omega(.)) = Sigma(n)(k=1) omega (k) X-k, n >= 1 where omega(t). 0 <= t <= 1 is a Lipschitz function of order 1 with parallel to omega(.)parallel to(2) = root integral(1)(0)omega(2)(t)dt > 0, necessary and sufficient conditions are provided for X to enjoy iterated logarithm type behavior of the form 0 < lim SUPn-->infinity |S-n(X, omega(.))/root nh(n) < infinity almost surely where h(.) is a positive, nondecreasing function which is slowly varying at infinity. Some corollaries are presented for particular choices of h(.).