arXiv subjects

Anne-Laure Basdevant

Publications and source records attributed to Anne-Laure Basdevant.

24 records · Page 2Linked to original sources

arXiv2007

Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent

We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it and group together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is the allele frequency spectrum. The celebrated Ewens sampling formula gives precise probabilities for the allele frequency spectrum associated with Kingman's coalescent. This (and the degenerate star-shaped coalescent) are the only Lambda coalescents for which explicit probabilities are known, although they are known to satisfy a recursion due to Moehle. Recently, Berestycki, Berestycki and Schweinsberg have proved asymptotic results for the allele frequency spectra of the Beta(2-alpha,alpha) coalescents with alpha in (1,2). In this paper, we prove full asymptotics for the case of the Bolthausen-Sznitman coalescent.

math.PR↗

arXiv2007

Rate of growth of a transient cookie random walk

We consider a one-dimensional transient cookie random walk. It is known from a previous paper that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $α>1$ or $\le 1$. In this article, we give the exact rate of growth of $(X_n)$ in the zero speed regime, namely: for $0<α<1$, $X_n/n^{\frac{α+1}{2}}$ converges in law to a Mittag-Leffler distribution whereas for $α=1$, $X_n(\log n)/n$ converges in probability to some positive constant.

math.PR↗

arXiv2006

On the speed of a cookie random walk

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just 3 cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.

math.PR↗

arXiv2006

On the equivalence of some eternal additive coalescents

In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

math.PR↗

arXiv2005

Fragmentation of compositions and intervals

The fragmentation processes of exchangeable partitions have already been studied by several authors. In this paper, we examine rather fragmentation of exchangeable compositions, that means partitions of $\mathbb{N}$ where the order of the blocks counts. We will prove that such a fragmentation is bijectively associated to an interval fragmentation. Using this correspondence, we then calculate the Hausdorff dimension of certain random closed set that arise in interval fragmentations and we study Ruelle's interval fragmentation.

math.PR↗

arXiv2005

Ruelle's probability cascades seen as a fragmentation process

In this paper, we study Ruelle's probability cascades in the framework of time-inhomogeneous fragmentation processes. We describe Ruelle's cascades mechanism exhibiting a family of measures $(ν_t,t\in [0,1[)$ that characterizes its infinitesimal evolution. To this end, we will first extend the time-homogeneous fragmentation theory to the inhomogeneous case. In the last section, we will study the behavior for small and large times of Ruelle's fragmentation process.

math.PR↗