AISM 54, 748-757

© 2002 ISM

## 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^{2}

^{1}UMR 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 $X_{1},\ldots,X_{n}$ 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.

**Source**
(TeX , DVI )