Erdos-Renyi Laws

Abstract

Almost sure limit theorems are proved for maxima of functions of moving blocks of size $c \log n$ of independent rv's and for maxima of functions of the empirical probability measures of these blocks. It is assumed that for the functions considered a first-order large deviation statement holds. It is well known that the indices of these large deviations are, in most cases, expressible in terms of Kullback-Leibler information numbers, and the a.s. limits of the above maxima are the inverses of these indices evaluated at $1/c$. Several examples are presented as corollaries for frequently used test statistics and point estimators.