Back to the Local Score in the Logarithmic Case: A Direct and Simple Proof

Let X₁, ... , X_n be a sequence of i.i.d. integer valued random variables and H_n 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 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.