Euclidean algorithms are Gaussian

Viviane Baladi (1) , Brigitte Vallée (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 Equipe AMACC - Laboratoire GREYC - UMR6072

This study provides new results about the probabilistic behaviour of a class of Euclidean algorithms: the asymptotic distribution of a whole class of cost-parameters associated to these algorithms is normal. For the cost corresponding to the number of steps Hensley already has proved a Local Limit Theorem; we give a new proof, and extend his result to other euclidean algorithms and to a large class of digit costs, obtaining a faster, optimal, rate of convergence. The paper is based on the dynamical systems methodology, and the main tool is the transfer operator. In particular, we use recent results of Dolgopyat.

Domaines: Algorithme et structure de données [cs.DS]; Complexité [cs.CC]

Viviane Baladi, Brigitte Vallée. Euclidean algorithms are Gaussian. 2004. ⟨hal-00012771⟩