R. Kiesel et U. Stadtmuller, ERDOS-RENYI-SHEPP LAWS AND WEIGHTED SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM-VARIABLES, Journal of theoretical probability, 9(4), 1996, pp. 961-982
Let X(0), X(1), X(2),... be i.i.d. random variables with E(X(0))=0, E(
X(0)(2))=1, E(exp{tX(0)}) < infinity (\t\ < t(0)) and partial sums S-n
. Starting from Shepp's version of the well-known Erdos-Renyl-Shepp la
w [GRAHICS] where alpha is a number depending upon c and the distribut
ion of X(0), we show that other weighted sums V(n)= Sigma a(j)(n) X(j)
exhibit a similar lim sup behavior, if the weights satisfy certain re
gularity conditions. We also prove for such weighted sums certain vers
ions of the classical Erdos-Renyi law.