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Jean Bertoin

Publications and source records attributed to Jean Bertoin.

At least 55 records · Page 3Linked to original sources
arXiv2013

The cut-tree of large Galton-Watson trees and the Brownian CRT

Consider the edge-deletion process in which the edges of some finite tree T are removed one after the other in the uniform random order. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this edge-deletion process. Our main result shows that after a proper rescaling, the cut-tree of a critical Galton-Watson tree with finite variance and conditioned to have size n, converges as $n\to\infty$ to a Brownian continuum random tree (CRT) in the weak sense induced by the Gromov-Prokhorov topology. This yields a multi-dimensional extension of a limit theorem due to Janson [Random Structures Algorithms 29 (2006) 139-179] for the number of random cuts needed to isolate the root in Galton-Watson trees conditioned by their sizes, and also generalizes a recent result [Ann. Inst. Henri Poincaré Probab. Stat. (2012) 48 909-921] obtained in the special case of Cayley trees.

math.PR↗
arXiv2013

Local times for functions with finite variation: two versions of Stieltjes change of variables formula

We introduce two natural notions for the occupation measure of a function $V$ with finite variation. The first yields a signed measure, and the second a positive measure. By comparing two versions of the change-of-variables formula, we show that both measures are absolutely continuous with respect to Lebesgue measure. Occupation densities can be thought of as local times of $V$, and are described by a Meyer-Tanaka like formula.

math.PR↗
arXiv2013

On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees

We consider a Bernoulli bond percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p_n=1-c/\ln n$ for some $c>0$ fixed. It is known that with high probability, there exists then a unique giant cluster of size $G_n\sim \e^{-c}$, and it follows from a recent result of Schweinsberg \cite{Sch} that $G_n$ has non-gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees.

math.PR↗
arXiv2013

Increasing processes and the change of variables formula for non-decreasing functions

Given an increasing process $(A_t)_{t\geq 0}$, we characterize the right-continuous non-decreasing functions $f: \R_+\to \R_+$ that map $A$ to a pure-jump process. As an example of application, we show for instance that functions with bounded variations belong to the domain of the extended generator of any subordinators with no drift and infinite Lévy measure.

math.PR↗
arXiv2013

Almost giant clusters for percolation on large trees with logarithmic heights

This text is based on a lecture for the Sheffield Probability Day; its main purpose is to survey some recent asymptotic results about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

math.PR↗
arXiv2012

Supercritical percolation on large scale-free random trees

We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest clusters, extending a recent result for large random recursive trees. The approach relies on the analysis of the asymptotic behavior of branching processes subject to rare neutral mutations, which may be of independent interest.

math.PR↗
arXiv2012

On largest offsprings in a critical branching process with finite variance

We continue our study of the distribution of the maximal number $X^{\ast}_k$ of offsprings amongst all individuals in a critical Galton-Watson process started with $k$ ancestors, treating the case when the reproduction law has a regularly varying tail $\bar F$ with index $-α$ for $α>2$ (and hence finite variance). We show that $X^{\ast}_k$ suitably normalized converges in distribution to a Frechet law with shape parameter $α/2$; this contrasts sharply with the case $1<α<2$ when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in the decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

math.PR↗
arXiv2012

Coagulation with limited aggregations

Smoluchowski's coagulation equations can be used as elementary mathematical models for the formation of polymers. We review here some recent contributions on a variation of this model in which the number of aggregations for each atom is a priori limited. Macroscopic results in the deterministic setting can be explained at the microscopic level by considering a version of stochastic coalescence with limited aggregations, which can be related to the so-called random configuration model of random graph theory.

math.PR↗
arXiv2011

Functional limit theorems for Lévy processes satisfying Cramér's condition

We consider a Lévy process that starts from $x<0$ and conditioned on having a positive maximum. When Cramér's condition holds, we provide two weak limit theorems as $x\to -\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.

math.PR↗
arXiv2011

The area of a self-similar fragmentation

We consider the area $A=\int_0^{\infty}\left(\sum_{i=1}^{\infty} X_i(t)\right) \d t$ of a self-similar fragmentation process $\X=(\X(t), t\geq 0)$ with negative index. We characterize the law of $A$ by an integro-differential equation. The latter may be viewed as the infinitesimal version of a recursive distribution equation that arises naturally in this setting. In the case of binary splitting, this yields a recursive formula for the entire moments of $A$ which generalizes known results for the area of the Brownian excursion.

math.PR↗
arXiv2010

Fires on trees

We consider random dynamics on the edges of a uniform Cayley tree with $n$ vertices, in which edges are either inflammable, fireproof, or burt. Every inflammable edge is replaced by a fireproof edge at unit rate, while fires start at smaller rate $n^{-α}$ on each inflammable edge, then propagate through the neighboring inflammable edges and are only stopped at fireproof edges. A vertex is called fireproof when all its adjacent edges are fireproof. We show that as $n\to \infty$, the density of fireproof vertices converges to $1$ when $α>1/2$, to $0$ when $α<1/2$, and to some non-degenerate random variable when $α=1/2$. We further study the connectivity of the fireproof forest, in particular the existence of a giant component.

math.PR↗
arXiv2010

Asymptotic regimes for the partition into colonies of a branching process with emigration

We consider a spatial branching process with emigration in which children either remain at the same site as their parents or migrate to new locations and then found their own colonies. We are interested in asymptotics of the partition of the total population into colonies for large populations with rare migrations. Under appropriate regimes, we establish weak convergence of the rescaled partition to some random measure that is constructed from the restriction of a Poisson point measure to a certain random region, and whose cumulant solves a simple integral equation.

math.PR↗
arXiv2010

A two-time-scale phenomenon in a fragmentation-coagulation process

Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$, or vice-versa (provided of course that $A$, or $B$, is not empty). The number of balls in $B$ after $n$ steps is of order $\sqrt n$, and this number remains essentially the same after $\sqrt n$ further steps. Observe that each ball in the urn $B$ after $n$ steps has a probability bounded away from $0$ and $1$ to be placed back in the urn $A$ after $\sqrt n$ further steps. So, even though the number of balls in $B$ does not evolve significantly between $n$ and $n+\sqrt n$, the precise contain of urn $B$ does. This elementary observation is the source of an interesting two-time-scale phenomenon which we illustrate using a simple model of fragmentation-coagulation. Inspired by Pitman's construction of coalescing random forests, we consider for every $n\in \N$ a uniform random tree with $n$ vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits subaging in the sense that when it is observed after $k$ steps in the regime $k\sim tn+s\sqrt n$ with $t>0$ fixed, it seems to reach a statistical equilibrium as $n\to\infty$; but different values of $t$ yield distinct pseudo-stationary distributions.

math.PR↗
arXiv2010

Retrieving information from subordination

We show that if $(X_s, s\geq 0)$ is a right-continuous process, $Y_t=\int_0^t\d s X_s$ its integral process and $τ= (τ_{\ell}, \ell \geq 0)$ a subordinator, then the time-changed process $(Y_{τ_{\ell}}, \ell\geq 0)$ allows to retrieve the information about $(X_{τ_{\ell}}, \ell\geq 0)$ when $τ$ is stable, but not when $τ$ is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable.

math.PR↗
arXiv2009

Some applications of duality for Lévy processes in a half-line

The central result of this paper is an analytic duality relation for real-valued Lévy processes killed upon exiting a half-line. By Nagasawa's theorem, this yields a remarkable time-reversal identity involving the Lévy process conditioned to stay positive. As examples of applications, we construct a version of the Lévy process indexed by the entire real line and started from $-\infty$ which enjoys a natural spatial-stationarity property, and point out that the latter leads to a natural Lamperti-type representation for self-similar Markov processes in $(0,\infty)$ started from the entrance point 0+.

math.PR↗
arXiv2009

The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations

We consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton-Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given.

math.PR↗
arXiv2009

A limit theorem for trees of alleles in branching processes with rare neutral mutations

We are interested in the genealogical structure of alleles for a Bienaymé-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jirina process) in discrete time. Itô's excursion theory and the Léevy-Itô decomposition of subordinators provide fundamental insights for the results.

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