ERDOS-RENYI-SHEPP LAWS AND WEIGHTED SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM-VARIABLES

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.