Consider two ancestral lineages sampled from a system of two-dimensional branching random walks with logistic regulation in the stationary regime. We study the asymptotics of their coalescence time for large initial separation and find that it agrees with well known results for a suitably scaled two-dimensional stepping stone model and also with Malécot's continuous-space approximation for the probability of identity by descent as a function of sampling distance.
This can be viewed as a justification for the replacement of locally fluctuating population sizes by fixed effective sizes. Our main tool is a joint regeneration construction for the spatial embeddings of the two ancestral lineages.[-]

Les chaînes de Markov à mémoire de longueur variable constituent une classe de sources probabilistes. Il sera question dans cet exposé d'existence et unicité de mesure invariante pour une collection d'exemples de chaînes. Nous nous intéresserons également au comportement asymptotique d'une marche aléatoire dont les longueurs de sauts ne sont pas forcément intégrables. Les lois de sauts dépendent partiellement du passé de la trajectoire. Plus précisément, la probabilité de monter ou de descendre dépend du temps passé dans la direction dans laquelle le marcheur est en train d'avancer. Un critère de récurrence/transience s'exprimant en fonction des paramètres du modèle sera énoncé. Suivront plusieurs exemples illustrant le caractère instable du type de la marche lorsqu'on perturbe légèrement les paramètres.
Les travaux décrits dans cet exposé ont été faits en collaboration avec B. Chauvin, F. Paccaut et N. Pouyanne ou B. de Loynes, A. Le Ny et Y. Offret.[-]

It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional limit theorem for the sine-process, which turns out to be very different from the Donsker Invariance Principle. We show that the anti-derivative of our process can be approximated by the sum of a linear Gaussian process and small independent Gaussian fluctuations whose covariance matrix we compute explicitly.[-]

We study Bessel and Dunkl processes $\left(X_{t, k}\right)_{t>0}$ on $\mathbb{R}^{N}$ with possibly multivariate coupling constants $k \geq 0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_{N}$ these Bessel processes are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types $\mathrm{A}$ and $\mathrm{B}$ and derive analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N \rightarrow \infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type $\mathrm{B}$ new non-symmetric semicircle-type limit distributions on $\mathbb{R}$ appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko-Pastur limit laws for the empirical measures of the zeroes of Hermite and Laguerre polynomials respectively. (based on joint work with Michael Voit)[-]

The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, weak convergence is proved for a sequence of Markov chains consisting of two components related to the Kingman coalescent, under a parent dependent d-alleles mutation scheme, as the sample size, grows to infinity. The first component is the normalised d-dimensional jump chain of the block counting processes of the Kingman coalescent. The second component is a d^2-dimensional process counting the number of mutations between types occurring in the Kingman coalescent. Time is scaled by the sample size. The limiting process consists of a deterministic d-dimensional component, describing the limit of the block counting jump chain, and d^2 independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. The weak convergence result is first proved, using a generator approach, in the setting of parent independent mutations. A change of measure argument is used to extend the weak convergence result to include parent dependent mutations.[-]

This talk will focus on the fluctuations of a linear spectral statistic around its mean for $P\left(W_N, D_N\right)$ where $P$ is a polynomial, $W_N$ a Wigner matrix and $D_N$ a deterministic diagonal matrix. I will first consider the case when $P\left(W_N,D_N\right)=W_N+D_N$, based on a joint work with M. Février (U. Paris-Saclay). In the general case of $P$ a selfadjoint noncommutative polynomial, I will present results for the special case of the Stieltjes transform, based on a joint work with S. Belinschi (CNRS, U. Toulouse), M. Capitaine (CNRS,U. Toulouse) and M. Février (U. Paris-Saclay).[-]

We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}F\left ( \xi \left ( n \right ),\xi \left ( 2n \right ),...,\xi \left ( \ell n \right ) \right )$ where $\xi \left ( n \right )$ is a stochastic process or a dynamical system. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple recurrence properties of corresponding stochastic processes and dynamical systems. Among our results are: central limit theorem, a.s. central limit theorem, local limit theorem, large deviations and averaging. Some multifractal type questions and open problems will be discussed, as well.
Keywords : limit theorems - nonconventional sums - multiple recurrence[-]

The Poisson limit theorem which appeared in 1837 seems to be the first law of rare events in probability. Various generalizations of it and estimates of errors of Poisson approximations were obtained in probability and more recently this became a popular topic in dynamics in the form of study of asymptotics of numbers of arrivals at small (shrinking) sets by a stochastic process or by a dynamical system. I will describe recent results on Poisson and compound Poisson asymptotics in a nonconventional setup, i.e. for numbers of events of multiple returns to shrinking sets, namely, for numbers of combined events of the type $\left \{ \omega : \xi \left ( jn,\omega\right )\in \Gamma_N,j = 1,...,\ell \right \},n\leq N$ where $\xi \left ( k,\omega \right )$ is defined as a stochastic process from the beginning or it is built from a dynamical system by writing $\xi \left ( k,\omega \right )=T^k\omega .$ We obtain an essentially complete description of possible limiting behaviors of distributions of numbers of multiple recurrencies to shrinking cylinders for $\psi $-mixing shifts. Some possible extensions and related questions will be discussed, as well. Most of the results were obtained jointly with my student Ariel Rapaport and some of them are new even for the widely studied single (conventional) recurrencies case.
Keywords : Poisson limit theorems - nonconventional sums - multiple recurrence[-]