Showing papers in "ALEA-Latin American Journal of Probability and Mathematical Statistics in 2008"

Limit theorems for an epidemic model on the com- plete graph

Abstract: We study the following random walks system on the complete graph with n vertices. At time zero, there is a number of active and inactive particles living on the vertices. Active particles move as continuous-time, rate 1, random walks on the graph, and, any time a vertex with an inactive particle on it is visited, this particle turns into active and starts an independent random walk. However, for a fixed integer L 1, each active particle dies at the instant it reaches a total of L jumps without activating any particle. We prove a Law of Large Numbers and a Central Limit Theorem for the proportion of visited vertices at the end of the process.

Limit theorems for multiple stochastic integrals

Abstract: We show that the general stable convergence results proved in Peccati and Taqqu (2007) for generalized adapted stochastic integrals can be used to ob- tain limit theorems for multiple stochastic integrals with respect to independently scattered random measures. Several applications are developed in a companion pa- per (see Peccati and Taqqu, 2008a), where we prove central limit results involving single and double Poisson integrals, as well as quadratic functionals associated with moving average Levy processes.

Random Walk in deterministically changing envi- ronment

Abstract: We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.

Convergence to the viscous porous medium equa- tion and propagation of chaos

Abstract: We study a sequence of nonlinear stochastic differential equations and show that the distributions of the solutions converge to the solution of the vis- cous porous medium equation with exponent m > 1, generalizing the results of Oelschlager (2001) and Philipowski (2006) which concern the case m = 2. Fur- thermore we explain how to apply this result to the study of interacting particle systems.

The critical contact process in a randomly evolving environment dies out

Abstract: Bezuidenhout and Grimmett proved that the critical contact process dies out. Here, we generalize the result to the so called contact process in a random evolving environment (CPREE), introduced by Erik Broman. This process is a generalization of the contact process where the recovery rate can vary between two values. The rate which it chooses is determined by a background process, which evolves independently at different sites. As for the contact process, we can similarly define a critical value in terms of survival for this process. In this paper we prove that this definition is independent of how we start the background process, that finite and infinite survival (meaning nontriviality of the upper invariant measure) are equivalent and finally that the process dies out at criticality.

Zonoids induced by Gauss measure with an application to risk aversion

Abstract: Suppose E is a real, separable Banach space and for each x ∈ E denote by co{0,(1, x)} the line segment joining the two points 0 and (1, x) in R × E. The aim of this paper is to discuss the strong law of large numbers and the central limit theorem for the random line segment co{0,(1, a + X)} when X is a centred Gaussian random vector in E and a ∈ E. Finally, an application to mathematical finance is given.

Fluctuations for a conservative interface model on a wall

Abstract: We consider an effective interface model on a hard wall in (1+1) di- mensions, with conservation of the area between the interface and the wall. We prove that the equilibrium fluctuations of the height variable converge in law to the solution of a SPDE with reflection and conservation of the space average. The proof is based on recent results obtained with L. Ambrosio and G. Savare on stability properties of Markov processes with log-concave invariant measures.