Nous considérons la fraction continue ordinaire de pour , ou, de manière équivalente, l’algorithme de pgcd d’Euclide pour deux entiers , avec grand et et distribués uniformément. Nous étudions la distribution du coût total de l’exécution de l’algorithme pour un coût additif sur l’ensemble des «digits» possibles, lorsque tend vers l’infini. Le théorème de la limite locale a été démontré par le deuxième auteur si est non réseau et satisfait une condition de croissance modérée. En imposant une condition diophantienne sur le coût, nous parvenons à contrôler la vitesse de convergence dans ce théorème de la limite locale. Pour cela nous utilisons des estimées obtenues par le premier auteur et Vallée, et nous adaptons à notre problème des bornes de Dolgopyat et Melbourne sur les opérateurs de transfert. Notre condition diophantienne est générique (par rapport à la mesure de Lebesgue). Pour des observables assez régulières (par rapport à la condition diophantienne), nous obtenons la vitesse optimale.
For large , we consider the ordinary continued fraction of with , or, equivalently, Euclid’s gcd algorithm for two integers , putting the uniform distribution on the set of and . We study the distribution of the total cost of execution of the algorithm for an additive cost function on the set of possible digits, asymptotically for . If is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. Introducing diophantine conditions on the cost, we are able to control the speed of convergence in the local limit theorem. We use previous estimates of the first author and Vallée, and we adapt to our setting bounds of Dolgopyat and Melbourne on transfer operators. Our diophantine condition is generic (with respect to Lebesgue measure). For smooth enough observables (depending on the diophantine condition) we attain the optimal speed.
Mots clés : euclidean algorithms, local limit theorem, diophantine condition, speed of convergence, transfer operator, continued fraction
@article{AIHPB_2008__44_4_749_0, author = {Baladi, Viviane and Hachemi, A{\"\i}cha}, title = {A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {749--770}, publisher = {Gauthier-Villars}, volume = {44}, number = {4}, year = {2008}, doi = {10.1214/07-AIHP140}, mrnumber = {2446296}, zbl = {1231.37015}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP140/} }
TY - JOUR AU - Baladi, Viviane AU - Hachemi, Aïcha TI - A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 749 EP - 770 VL - 44 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP140/ DO - 10.1214/07-AIHP140 LA - en ID - AIHPB_2008__44_4_749_0 ER -
%0 Journal Article %A Baladi, Viviane %A Hachemi, Aïcha %T A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 749-770 %V 44 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP140/ %R 10.1214/07-AIHP140 %G en %F AIHPB_2008__44_4_749_0
Baladi, Viviane; Hachemi, Aïcha. A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 749-770. doi : 10.1214/07-AIHP140. http://www.numdam.org/articles/10.1214/07-AIHP140/
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