Precise asymptotics in complete moment convergence for self-normalized sums
Let X, X1, X2.... be a sequence of nondegenerate i.i.d. random variables with zero means, set S-n = X-1 +...+ X-n and V-n(2) = X-1(2) +... +X-n(2), (EXI)-I-2 vertical bar X vertical bar   2, delta > 2/ beta-1, lim(epsilon down arrow 0)epsilon(beta(delta+1)-2)Sigma(infinity)(n=2) (log n)(delta-2/beta)/n E(S-n/(Vn)I)-I-2(vertical bar S-n vertical bar >= epsilon V(n()log n)(1/beta)) = beta E vertical bar N vertical bar(beta(delta+1))/beta(delta+1)-2, where N is a standard normal random variable. (C) 2008 Elsevier Ltd. All rights reserved.
precise asymptotics complete moment convergence self-normalized sums