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Benaim, Michel
Nom
Benaim, Michel
Affiliation principale
Fonction
Professeur ordinaire
Email
michel.benaim@unine.ch
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Résultat de la recherche
Voici les éléments 1 - 5 sur 5
- PublicationAccès libreA class of mean field interaction models for computer and communication systems(2008)
; Le Boudec, Jean-YvesWe consider models of N interacting objects, where the interaction is via a common resource and the distribution of states of all objects. We introduce the key scaling concept of intensity; informally, the expected number of transitions per object per time slot is of the order of the intensity. We consider the case of vanishing intensity, i.e. the expected number of object transitions per time slot is o(N). We show that, under mild assumptions and for large N, the occupancy measure converges, in mean square (and thus in probability) over any finite horizon, to a deterministic dynamical system. The mild assumption is essentially that the coefficient of variation of the number of object transitions per time slot remains bounded with N. No independence assumption is needed anywhere. The convergence results allow us to derive properties valid in the stationary regime. We discuss when one can assure that a stationary point of the ODE is the large N limit of the stationary probability distribution of the state of one object for the system with N objects. We use this to develop a critique of the fixed point method sometimes used in conjunction with the decoupling assumption. (C) 2008 Elsevier B.V. All rights reserved. - PublicationAccès libreStochastic approximations and differential inclusions(2005)
; - PublicationAccès libreDeterministic approximation of stochastic evolution in games(2003)
; Weibull, Jorgen WThis paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The processes are Markov chains, and the approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the discrete stochastic process, for large populations, and its deterministic flow approximation. In particular, we provide probabilistic bounds on exit times from and visitation rates to neighborhoods of attractors; to the deterministic flow. We sharpen these results in the special case of ergodic processes. - PublicationAccès libreVertex-reinforced random walks and a conjecture of pemantle(1997)We discuss and disprove a conjecture of Pemantle concerning vertex-reinforced random walks. The setting is a general theory of non-Markovian discrete-time random processes on a finite space E = {1,...,d}, for which the transition probabilities at each step are influenced by the proportion of times each state has been visited. It is shown that, under mild conditions, the asymptotic behavior of the empirical occupation measure of the process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the d - 1 unit simplex. In particular, any minimal attractor of this vector field has a positive probability to be the Limit set of the sequence of empirical occupation measures. These properties are used to disprove a conjecture and to extend some results due to Pemantle. Some applications to edge-reinforced random walks are also considered.
- PublicationAccès libreA dynamical system approach to stochastic approximations(1996)It is known that some problems of almost sure convergence for stochastic approximation processes can be analyzed via an ordinary differential equation (ODE) obtained by suitable averaging. The goal of this paper is to show that the asymptotic behavior of such a process can be related to the asymptotic behavior of the ODE without any particular assumption concerning the dynamics of this ODE. The main results are as follows: a) The limit sets of trajectory solutions to the stochastic approximation recursion are, under classical assumptions, almost surely nonempty compact connected sets invariant under the how of the ODE and contained in its set of chain-recurrence. b) If the gain parameter goes to zero at a suitable rate depending on the expansion rate of the ODE, any trajectory solution to the recursion is almost surely asymptotic to a forward trajectory solution to the ODE.