Precise asymptotics in the law of the iterated logarithm

Let X-1, X-2,... be id.d. random variables with EX1 = 0 and positive, finite variance sigma(2), and set S-n = X-1 + ... + X-n. For any alpha > -1, beta > -1/2 and for kappa(n)(epsilon) a function of epsilon and n such that kappa(n)(epsilon) log log n -> lambda as n up arrow infinity and epsilon down arrow root alpha + 1, EX12 (log vertical bar X-1 vertical bar)(alpha+1) (log log vertical bar X-1 vertical bar)(beta+1) < infinity, we prove that (lim)(epsilon down arrow root alpha+1) (epsilon(2) - (alpha + 1))(beta+1/2) (Sigma)(n > 3) n/(log n)(alpha)(log log n)(beta) P(vertical bar S-n vertical bar >= sigma root 2n log log n (epsilon + kappa(n)(epsilon))) = (1/root pi(alpha + 1)(-1/2) exp(-2 lambda root alpha + 1)Gamma(beta + 1/2).