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- Nicolas Pouyanne
- 2006

Pólya processes are natural generalization of Pólya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small… (More)

A " hybrid method " , dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit… (More)

- Brigitte Chauvin, Nicolas Pouyanne, Reda Sahnoun
- 2009

We consider a two colors Pólya urn with balance S. Assume it is a large urn i.e. the second eigenvalue m of the replacement matrix satisfies 1/2 < m/S ≤ 1. After n drawings, the composition vector has asymptotically a first deterministic term of order n and a second random term of order n m/S. The object of interest is the limit distribution of this random… (More)

- Nicolas Pouyanne
- 2005

This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having " large " eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn's replacement matrix and examples… (More)

Let m ≥ 3 be an integer. The so-called m-ary search tree is a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when m ≤ 26, the asymptotic behavior of the process is Gaussian, but for m ≥ 27 it is no longer… (More)

Let m be a positive integer, and p n (m) the proportion of permutations of the symmetric group S n that admit an m-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula p n (m) ∼ n→+∞ π m n 1−ϕ(m)/m , where ϕ is the Euler function and π m an explicit constant.

Pólya processes are natural generalization of Pólya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small… (More)

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical… (More)

Common assumptions on the source producing the words inserted in a suffix trie with n leaves lead to a ln n height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of n and another one whose saturation level is negligible with respect to ln n. Both are built from VLMC (Variable Length Markov Chain)… (More)