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            <title xml:lang="en">Statistical investigation of biological sequences and convergence of martingales</title>
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                <authority type="jury">Guy Fayolle (directeur)</authority>
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                <authority type="jury">Fabrice Gamboa (examinateur)</authority>
                <authority type="jury">Didier Piau (rapporteur)</authority>
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                <term xml:lang="en">strong laws</term>
                <term xml:lang="en">insertion depth</term>
                <term xml:lang="en">digital search tree</term>
                <term xml:lang="en">&lt;br /&gt;random tree</term>
                <term xml:lang="en">genomic signature</term>
                <term xml:lang="en">chi-squared test</term>
                <term xml:lang="en">iterated function system</term>
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                <term xml:lang="fr">test du chi-deux</term>
                <term xml:lang="fr">signature&lt;br /&gt;génomique</term>
                <term xml:lang="fr">arbre aléatoire</term>
                <term xml:lang="fr">arbre digital de recherche</term>
                <term xml:lang="fr">profondeur&lt;br /&gt;d'insertion</term>
                <term xml:lang="fr">martingales</term>
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              <p>The Chaos Game Representation is a dynamical system&lt;br /&gt;which maps a sequence of letters taken from a finite alphabet onto an empirical measure on a set. We show how the CGR can be used to&lt;br /&gt;characterize the&lt;br /&gt;order of an homogeneous Markov chain and to define a new family of tests.&lt;br /&gt;Then we propose a construction of Digital Search Trees, inspired&lt;br /&gt;from the CGR, by successively inserting all the returned prefixes of a Markov&lt;br /&gt;chain. We give the asymptotic behavior of the critical lengths of paths, which&lt;br /&gt;turns out to be, at first order, the same one as in the case of DST built from&lt;br /&gt;independent Markov chains.&lt;br /&gt;A last part deals with properties of almost sure convergence of vectorial&lt;br /&gt;martingales. Under suitable regularity conditions on the&lt;br /&gt;growing process, we establish the convergence of normalized moments of all&lt;br /&gt;orders in the almost sure central limit theorem. The results are&lt;br /&gt;applied to the cumulated errors of estimation and&lt;br /&gt;prediction in linear regression models and branching processes.</p>
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              <p>Le système dynamique Chaos Game Representation associe une suite de lettres dans un alphabet fini, une mesure empirique sur un ensemble. Fournit-elle plus d'information&lt;br /&gt;que les méthodes de comptage de mots classiques ? A&lt;br /&gt;partir d'une caractérisation basée sur la CGR, on propose une nouvelle famille de&lt;br /&gt;tests donnant l'ordre d'une chaîne de Markov homogène.&lt;br /&gt;On définit ensuite une construction d'arbres digitaux de recherche,&lt;br /&gt;inspirés par la CGR, en insérant successivement les préfixes retournés d'une chaîne de Markov. On montre que les longueurs des branches critiques se comportent, au premier ordre, comme si les&lt;br /&gt;séquences insérées étaient indépendantes entre elles.&lt;br /&gt;La dernière partie est consacrée à l'étude de la convergence presque sûre des moments normalisés de tout ordre de martingales vectorielles dans le théorème de la limite centrale&lt;br /&gt;presque sûr. Les résultats sont appliqués aux erreurs d'estimation et de prédiction dans les régressions linéaires et les processus de branchement.</p>
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