Abstract
Les X1,..., Xn be a sequence of i.i.d. integer valued random variables and Hn the local score of a sequence. A recent result shows that Hn is actually the maximum of an integer valued Lindley process. Therefore known results about the asymptotics distribution of the maximum of a weakly dependent process, give readily the expected result about the asymptotic behavior of the local score in the logarithmic case, with a simple way for computing the needed constants. Genomic sequence scoring is one of the most important applications of the local score. An example of an application of the local score on protein sequences is therefore given in the paper.