Random sums of random vectors and multitype families of productive individuals.
Random trees and applications.
Random Walks and Trees
These notes provide an elementary and self-contained introduction to branching random walks. Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to the main part, Section 3, where branching random walks...
Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive.
Rate of growth of a transient cookie random walk.
Recurrence and transience of a multi-excited random walk on a regular tree.
Recurrence for branching Markov chains.
Remarks on ergodicity and invariant occupation measure in branching diffusions with immigration
Renewal property of the extrema and tree property of the excursion of a one-dimensional brownian motion
Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
We introduce the notion of a restricted exchangeable partition of . We obtain integral representations, consider associated fragmentations, embeddings into continuum random trees and convergence to such limit trees. In particular, we deduce from the general theory developed here a limit result conjectured previously for Ford’s alpha model and its extension, the alpha-gamma model, where restricted exchangeability arises naturally.
Riccati's differential equation in birth-death processes.
This note reviews the occurrence of Riccati's equation in three birth-death type processes, and outlines their solutions.
Robust Parametric Estimation of Branching Processes with a Random Number of Ancestors
2000 Mathematics Subject Classification: 60J80.The paper deals with a robust parametric estimation in branching processes {Zt(n)} having a random number of ancestors Z0(n) as both n and t tend to infinity (and thus Z0(n) in some sense). The offspring distribution is considered to belong to a discrete analogue of the exponential family – the class of the power series offspring distributions. Robust estimators, based on one and several sample paths, are proposed and studied for all values of the offspring...
Small positive values for supercritical branching processes in random environment
Branching Processes in Random Environment (BPREs) are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of , as . More precisely, we characterize the exponential...
Smoluchowski's coagulation equation : probabilistic interpretation of solutions for constant, additive and multiplicative kernels
Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes
Some comments on quasi-birth-and-death processes and matrix measures.
Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems.
Some Notes on the Non-central Negative Binomial Distribution.
Some properties of superprocesses under a stochastic flow
For a superprocess under a stochastic flow in one dimension, we prove that it has a density with respect to the Lebesgue measure. A stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov’s Lp-theory for linear SPDE.
Some remarks on P. Levy's theorem for a net of discrete measures and the purity law on locally compact semigroups.