arXiv subjects

Jean Bertoin

Publications and source records attributed to Jean Bertoin.

At least 19 recordsLinked to original sources

arXiv2026

Another solvable population dynamic with uniform ancestral memory

We study a population model with non-overlapping generations, where individuals have a type and retain the memory of the types of all their ancestors. We assume that for each individual, the reproduction law depends on the empirical distribution of ancestral types, unlike for a classical multi-type branching process where it hinges on the parental type only. We further assume the mean reproduction matrix has rank one. Building on techniques from previous works with Bastien Mallein, which themselves draw on foundational contributions by Philippe Flajolet and collaborators in the context of analytic urns, we analyze the population at large generations. In particular, we determine the Malthusian parameter and estimate the expected size of the sub-population whose empirical distribution of ancestral types lies within a given set.

math.PR↗

arXiv2026

Preferential relocations enhance survival for Markov chains with killing

We investigate the impact on survival of a modification of the evolution of a sub-stochastic Markov chain that involves random relocations at previously visited states. Our central result is that such preferential relocations increase the persistence rate, meaning the survival probability decays more slowly than for the benchmark chain without relocations, and this improvement is strict under mild assumptions. We derive explicit lower bounds on the ratio of persistence rates when the relocation distribution is highly dispersed. The analysis relies on ergodic properties of so-called chains with complete connections, and a Feynman--Kac approach to estimate persistence.

math.PR↗

arXiv2026

Local times and excursions for self-similar Markov trees

This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time measures $L(x,dt)$ at every level $x>0$ of the decoration for a large class of self-similar Markov trees. This enables us to mark at random a typical point in the tree at which the decoration is $x$. We identify the law of the decoration along the branch from the root to this tagged point in terms of a remarkable (positive) self-similar Markov process. We also show that after a proper normalization, $L(x,dt)$ converges as $x\to 0+$ to the harmonic measure $\mu$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $\lambda$ to compute distances on the tree turn the latter into a continuous branching tree. This is relevant to analyze the excusions of the decoration away from a given level. Many results of the present work shall be compared with the recent ones in [22,23] about local times and excursions of a Markov process indexed by L\'evy tree.

math.PR↗

arXiv2025

Reinforced Galton--Watson processes III: Empirical offspring distributions

Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution depends on the empirical offspring distribution of each individual along its ancestral lineage. We are interested here in asymptotic properties of the empirical distributions observed in the population, such as concentration, evanescence and persistence. For this, we incorporate tools from the theory of large deviations to our preceding analysis [arXiv:2306.02476,arXiv:2310.19030].

math.PR↗

arXiv2024

On a population model with memory

Consider first a memoryless population model described by the usual branching process with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a modification of the dynamics where individuals keep full memory of their forebears and procreation involves the reactivation of a gene picked at random on the ancestral lineage. By comparing the spectral radii of the two mean reproduction matrices (with and without memory), we observe that, on average, the model with memory always grows at least as fast as the model without memory. The proof relies on analyzing a biased Markov chain on the space of memories, and the existence of a unique ergodic law is demonstrated through asymptotic coupling.

math.PR↗

arXiv2024

Self-similar Markov trees and scaling limits

Self-similar Markov trees constitute a remarkable family of random compact real trees carrying a decoration function that is positive on the skeleton. As the terminology suggests, they are self-similar objects that further satisfy a Markov branching property. They are built from the combination of the recursive construction of real trees by gluing line segments with the seminal observation of Lamperti, which relates positive self-similar Markov processes and Levy processes via a time change. They carry natural length and harmonic measures, which can be used to perform explicit spinal decompositions. Self-similar Markov trees encompass a large variety of random real trees that have been studied over the last decades, such as the Brownian CRT, stable Levy trees, fragmentation trees, and growth-fragmentation trees. We establish general invariance principles for Galton--Watson trees with integer types and illustrate them with many combinatorial classes of random trees that have been studied in the literature.

math.PR↗

arXiv2023

Reinforced Galton-Watson processes I: Malthusian exponents

In a reinforced Galton-Watson process with reproduction law $\boldsymbolν$ and memory parameter $q\in(0,1)$, the number of children of a typical individual either, with probability $q$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability $1-q$, is given by an independent sample from $\boldsymbolν$. We estimate the average size of the population at a large generation, and in particular, we determine explicitly the Malthusian growth rate in terms of $\boldsymbolν$ and $q$. Our approach via the analysis of transport equations owns much to works by Flajolet and co-authors.

math.PR↗

arXiv2023

Reinforced Galton-Watson processes II: Large time behaviors

Reinforced Galton-Watson processes have been introduced in arxiv:2306.02476 as population models with non-overlapping generations, such that reproduction events along genealogical lines can be repeated at random. We investigate here some of their sample path properties such as asymptotic growth rates and survival, for which the effects of reinforcement on the evolution appear quite strikingly.

math.PR↗

arXiv2022

Limits of Pólya urns with innovations

We consider a version of the classical Pólya urn scheme which incorporates innovations. The space $S$ of colors is an arbitrary measurable set. After each sampling of a ball in the urn, one returns $C$ balls of the same color and additional balls of different colors given by some finite point process $ξ$ on $S$. When the number of steps goes to infinity, the empirical distribution of the colors in the urn converges to the normalized intensity measure of $ξ$, and we analyze the fluctuations. The ratio $ρ= E(C)/E(R)$ of the average number of copies to the average total number of balls returned plays a key role.

math.PR↗

arXiv2022

A model for an epidemic with contact tracing and cluster isolation, and a detection paradox

We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation. This enables us to apply general limit theorems for super-critical Crump-Mode-Jagers branching processes. Notably, we compute explicitly the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic. Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection; and we explain the reasons behind this seeming paradox.

math.PR↗

arXiv2022

Counterbalancing steps at random in a random walk

A random walk with counterbalanced steps is a process of partial sums $\check S(n)=\check X_1+ \cdots + \check X_n$ whose steps $\check X_n$ are given recursively as follows. For each $n\geq 2$, with a fixed probability $p$, $\check X_n$ is a new independent sample from some fixed law $μ$, and with complementary probability $1-p$, $\check X_n= -\check X_{v(n)}$ counterbalances a previous step, with $v(n)$ a uniform random pick from $\{1, \ldots, n-1\}$. We determine the asymptotic behavior of $\check S(n)$ in terms of $p$ and the first two moments of $μ$. Our approach relies on a coupling with a reinforcement algorithm due to H.A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution $μ$ belongs to the domain of attraction of a stable law.

math.PR↗

arXiv2022

Counting the zeros of an elephant random walk

We study how memory impacts passages at the origin for a so-called elephant random walk in the diffusive regime. We observe that the number of zeros always grows asymptotically like the square root of the time, despite the fact that, depending on the memory parameter, first return times to $0$ may have a finite expectation or a fat tail with exponent less than $1/2$. We resolve this apparent paradox by recasting the questions in the framework of scaling limits for Markov chains and self-similar Markov processes.

math.PR↗

arXiv2021

Scaling limits of branching random walks and branching stable processes

Branching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanism for point processes. Here, we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results that have been obtained during the last decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.

math.PR↗

arXiv2021

On the local times of noise reinforced Bessel processes

We investigate the effects of noise reinforcement on a Bessel process of dimension $d\in(0,2)$, and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

math.PR↗

arXiv2020

How linear reinforcement affects Donsker's Theorem for empirical processes

A reinforcement algorithm introduced by H.A. Simon \cite{Simon} produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability $p\in(0,1)$, $\hat U_{n+1}$ is sampled uniformly from $\hat U_1, \ldots, \hat U_n$, and with complementary probability $1-p$, $\hat U_{n+1}$ is a new independent uniform variable. The Glivenko-Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $p<1/2$, and that a further rescaling is needed when $p>1/2$ and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

math.PR↗

arXiv2020

Universality of Noise Reinforced Brownian Motions

A noise reinforced Brownian motion is a centered Gaussian process $\hat B=(\hat B(t))_{t\geq 0}$ with covariance $E(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t,$ where $p\in(0,1/2)$ is a reinforcement parameter. Our main purpose is to establish a version of Donsker's invariance principle for a large family of step-reinforced random walks in the diffusive regime, and more specifically, to show that $\hat B$ arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.

math.PR↗

arXiv2020

On a two-parameter Yule-Simon distribution

We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in Bertoin [2019] in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in the mentioned paper. Finally, by superposing mutations to the branching process, we propose a model which leads to the full two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon [1955] on limiting word frequencies.

math.PR↗

arXiv2020

Elephant Random Walks and their connection to Pólya-type urns

In this paper, we explain the connection between the Elephant Random Walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced by Schütz and Trimper [2004] to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a parameter $p$ between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of $p$. Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.

cond-mat.stat-mech↗