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Bénédicte Haas

Publications and source records attributed to Bénédicte Haas.

24 records · Page 2Linked to original sources

arXiv2011

Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

We discuss the existence and characterization of quasi-stationary distributions and Yaglom limits of self-similar Markov processes that reach 0 in finite time. By Yaglom limit, we mean the existence of a deterministic function $g$ and a non-trivial probability measure $ν$ such that the process rescaled by $g$ and conditioned on non-extinction converges in distribution towards $ν$. If the study of quasi-stationary distributions is easy and follows mainly from a previous result by Bertoin and Yor \cite{BYFacExp} and Berg \cite{bergI}, that of Yaglom limits is more challenging. We will see that a Yaglom limit exits if and only if the extinction time at 0 of the process is in the domain of attraction of an extreme law and we will then treat separately three cases, according whether the extinction time is in the domain of attraction of a Gumbel law, a Weibull law or a Fréchet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Lévy process are given for the extinction time to be in the required domain of attraction. The limit of the process conditioned to be positive is then characterized by a multiplicative equation which is connected to a factorization of the exponential distribution in the Gumbel case, a factorization of a Beta distribution in the Weibull case and a factorization of a Pareto distribution in the Fréchet case. This approach relies partly on results on the tail distribution of the extinction time, which is known to be distributed as the exponential integral of a Lévy process. In that aim, new results on such tail distributions are given, which may be of independent interest. Last, we present applications of the Fréchet case to a family of Ornstein-Uhlenbeck processes.

math.PR↗

arXiv2010

Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes

The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation. We note that two parameters significantly influence these large time behaviors: the rate of formation of "nearly-1 relative masses" (this rate is related to the behavior near 0 of the Lévy measure associated with the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a nontrivial limit which is related to the quasi-stationary solutions of the equation. Besides, these quasi-stationary solutions, or, equivalently, the quasi-stationary distributions of the self-similar Markov processes, are fully described.

math.PR↗

arXiv2009

Spinal partitions and invariance under re-rooting of continuum random trees

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson--Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

math.PR↗

arXiv2009

Behavior near the extinction time in self-similar fragmentations I: the stable case

The stable fragmentation with index of self-similarity $α\in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + α)^{-1}$--stable continuum random tree below height $t$, for $t \geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t \geq 0)$, as it approaches its time of extinction, $ζ$. In particular, we show that $t^{1/α}F((ζ- t)^+)$ converges in distribution as $t \to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $ζ$.

math.PR↗

arXiv2008

Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous's beta-splitting models and Ford's alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.

math.PR↗

arXiv2005

Fragmentation processes with an initial mass converging to infinity

We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let $F\_{1}^{(m)}(t),F\_{2}^{(m)}(t),...$ denote the decreasing rearrangement of the masses present at time $t$ in a such process, starting from an initial mass $m$. Let then $m\to \infty $. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence $(F\_{2}^{(m)},F\_{3}^{(m)},...)$ converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of $m-F\_{1}^{(m)}$ to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to $\infty $ will converge to a tree with a spine coding a fragmentation with immigration.

math.PR↗