Precise asymptotics in the law of the iterated logarithm

Let X, X-1, X-2,... be i.i.d. random variables with mean 0 and positive, fi nite variance sigma (2), and set S-n = X-1 + ... + X-n, n greater than or e qual to 1. Continuing earlier work related to strong laws, we prove the fol lowing analogs for the law of the iterated logarithm: lim(epsilon down arrow sigma root2) root epsilon (2)-2 sigma (2) Sigma (n g reater than or equal to3) 1/n P(\S-n\ greater than or equal to epsilon root n log log n + a(n)) = sigma root2 whenever a(n) = O(rootn(log log n)(-gamma)) for some gamma greater than or equal to 1/2 (assuming slightly more than finite variance), and lim(epsilon down arrow0)epsilon (2) Sigma (n greater than or equal to3) 1/n log n P(\S-n\ greater than or equal to epsilon rootn log log n) = sigma (2 ).