Back to the local score in the logarithmic case : a direct and simple proof

Jean-Noel BACRO, Jean-Jacques DAUDIN, Sabine MERCIER, Stéphane ROBIN

Ann. Inst. Statist. Math, 2002.

Abstract

Let X₁, ... X_n be a sequence of i.i.d. positive or negative integer valued random variables and H_n = max_{0 <= i <= j <= n}(X_i + ... + X_j) the local score of the sequence. A recent result shows that H_n is actually the maximum of an integer valued Lindley process. Therefore known results about the asymptotic distribution of the maximum of a weakly dependent process give readily the expected result about the asymptotic behavior of the local score, with a simple way for computing the needed constants.