The comparison of two biological sequences is an important tool for the analysis of data from molecular biology. In order to make such comparisons, assignments, called scores, are attributed to the different couples of components of the sequences (nucleotides or amino acids) and we search the regions that correspond to the maximal score, the local score.

The statistical problem is to test if the calculated score is significant or not, in order to highlight a biological link between the sequences. The goal of this thesis consists in studying the distribution of the local score.

For this, the sequences are represented by independent and identically distributed and integer valued random variables.

We first suppose that the expected score is non positive. The work of Karlin et al., implemented in B.L.A.S.T. for sequence alignment, is proved more precisely and ameliorated. Using the random walks theory, we then establish the distribution of the maximum of the partial sums. This distribution is the unique invariant probability transition of a Markov chain. This result allows us to obtain a new assymptotic (for long sequences) approximation for the local score distribution that improve the one given by Karlin et al.

We then obtain the exact distribution of the local score using Markov chain theory. This result is available for negative, positive or null expected score, and give the distribution as a power of a certain matrix and is adapted to short sequences.

These two approaches studied in the thesis are different and independent each other, and with the one used by Karlin et al. as well. The results can easily be generalized for Markovian dependent sequences.