Precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables

Let {X, X-n, n >= 1) be a sequence of i.i.d. random variables and set S-n = Sigma(n)(i=1) X-j. N is the standard normal random variable, then for d > 0 and beta > 0, we show that lim(epsilon SE arrow 0) epsilon(2 beta/d) Sigma(infinity)(n=3) (log log n)(beta-1)/n(3/2) log n E{|S-n| - epsilon sigma root n(log log n)(d/2)}+ = d sigma E|N|(2 beta/d+1/)beta(2 beta + d) holds if and only if EX = 0, EX2 = sigma(2). (c) 2012 Elsevier B.V. All rights reserved.