BACK TO THE LOCAL SCORE IN THE LOGARITHMIC CASE: A DIRECT AND SIMPLE PROOF

J.-N. BACRO^{1}, J.-J. DAUDIN^{1}, S. MERCIER^{2} AND S. ROBIN^{1}

1UMR INAPG/INRA 518, 16, rue Cl. Bernard, 75231, Paris Cedex05, France

2 Departement Mathematique et Informatique, Université de Toulouse II, 5 allées A. Machado, 31058 Toulouse, France

(Received March 26, 2001; revised September 17, 2001)

Abstract. Let X_{1}, …, 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.

Key words and phrases: Extremal index, genomic sequence, Lindley process, local score, Markov chain.