We prove a strong law of large numbers for moving averages of independent, identically distributed random variables with certain subexponential distributions. These random variables show a behavior that can be considered intermediate between the classical strong law and the Erdös-Rényi law. We further show that the difference from the classical behavior is due to the influence of extreme terms.

We study the asymptotic behavior of the empirical process when theunderlying data are Gaussian and exhibit seasonallong-memory. We prove that the limiting process can be quitedifferent from the limit obtained in the case of regularlong-memory. However, in both cases, the limiting process isdegenerated. We apply our results to von–Mises functionals andU-Statistics.

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministicmarkov process in infinite dimensions. Adv. Appl. Probab. 44(2012) 749–773], the authors addressed the question of averaging for a slow-fastPiecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the presentpaper, we carry on and complete this work by the mathematical analysis of the fluctuationsof the slow-fast system around the averaged limit. A central limit theorem is derived andthe associated Langevin approximation is considered. The motivation for this work is thestudy of stochastic conductance based neuron models which describe the propagation of anaction potential along a nerve fiber.

We consider a generalization of the so-called divide andcolor model recently introduced by Häggström. We investigate thebehavior of the magnetization in large boxes of the lattice $\mathbb{Z}^d$ and its fluctuations. Thus, Laws of Large Numbers and CentralLimit Theorems are proved, both quenched and annealed. We showthat the properties of the underlying percolation process deeplyinfluence the behavior of the coloring model. In the subcriticalcase, the limit magnetization is deterministic and the CentralLimit Theorem admits a Gaussian limit. Conversely, the limitmagnetization is not deterministic in the supercritical case andthe limit of the Central Limit Theorem is not Gaussian, except inthe particular model with exactly two colors which are equallyprobable. We also prove a Central Limit Theorem for the size of the intersection of the infinite cluster with large boxes in supercritical bond percolation.

In this article we prove new results concerning thestructure and the stability properties of the global attractor associatedwith a class of nonlinear parabolic stochastic partial differential equationsdriven by a standard multidimensional Brownian motion.We first use monotonicity methodsto prove that the random fields either stabilize exponentially rapidly withprobability one around one of the two equilibrium states, or that they set outto oscillate between them. In the first case we can also compute exactly thecorresponding Lyapunov exponents.The last case of our analysis reveals a phenomenon of exchange of stabilitybetween the two components of the global attractor. In order to prove thisasymptotic property, we show an exponential decay estimate between the randomfield and its spatial average under an additional uniform ellipticityhypothesis.

We consider an infinite system of hard balls in $\xR^d$ undergoing Brownian motionsand submitted to a smooth pair potential.It is modelized by an infinite-dimensional stochastic differential equationwith an infinite-dimensional local time term.Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also showthat Gibbs measures are reversible measures.

Fine regularity of stochastic processes is usually measured in a local way by localHölder exponents and in a global way by fractal dimensions. In the case of multiparameterGaussian random fields, Adler proved that these two concepts are connected under theassumption of increment stationarity property. The aim of this paper is to consider thecase of Gaussian fields without any stationarity condition. More precisely, we prove thatalmost surely the Hausdorff dimensions of the range and the graph in any ballB(t0,ρ)are bounded from above using the local Hölder exponent at t0. We definethe deterministic local sub-exponent of Gaussian processes, which allows to obtain analmost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of thesample path on an open interval are controlled almost surely by the minimum of the localexponents. Then, we apply these generic results to the cases of the set-indexed fractionalBrownian motion on RN, the multifractionalBrownian motion whose regularity function H is irregular and the generalized Weierstrassfunction, whose Hausdorff dimensions were unknown so far.

Let (Sn)n≥0 be a $\mathbb Z$ -random walk and $(\xi_{x})_{x\in \mathbb Z}$ be a sequence of independent andidentically distributed $\mathbb R$ -valued random variables,independent of the random walk. Let h be a measurable, symmetricfunction defined on $\mathbb R^2$ with values in $\mathbb R$ . We study theweak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$ , withvalues in D[0,1] the set of right continuous real-valuedfunctions with left limits, defined by \[ \sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1].\] Statistical applications are presented, in particular we prove a strong law of large numbersfor U-statistics indexed by a one-dimensional random walk using a result of [1].

Branching process approximation to the initial stages of an epidemicprocess has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems.One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, insuch a way that their paths coincide for as long as possible. Inthis paper, it is shown, in the context of a Markovian model of parasiticinfection, that coincidence can be achieved with asymptotically high probability until MN infections have occurred, as long asMN = o(N 2/3), where N denotes the total number of hosts.

For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which $\lim_{n\to\infty}(\sum_{i=1}^nX_i)/b_n=0$ almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]. The sharpness of the results is illustrated by examples.

In this paper two characterizations of the Pólyadistribution are obtained when its contagion parameter isnegative. One of them is based on mixtures and the other one isobtained by characterizing a subfamily of the discrete Pearsonsystem.

We investigate the steady state behavior of an M/G/1 queue with modified Bernoulli schedule server vacations. Batches of variable size arrive at the system according to a compound Poisson process. However, all arriving batches are not allowed into the system. The restriction policy differs when the server is available in the system and when he is on vacation. We obtain in closed form, the steady state probability generating functions for the number of customers in the queue for various states of the server, the average number of customers as well as their average waiting time in the queue and the system. Many special cases of interest including complete admissibility, partial admissibility and no server vacations have been discusssed. Some known results are derived as particular cases of our model.

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measureunder various assumptions on finite volume functional inequalities. We follow Royer's approach (Royer, 1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box [-n,n]d (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincaré. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x)){\rm d}t+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients f and gk,driven by a sequence (βk)k of i.i.d. fractional Brownianmotions of index H>1/2. Using the Malliavin calculus techniquesand a p-th moment maximal inequality for the infinite sum ofSkorohod integrals with respect to (βk)k, we prove that theequation has a unique solution (in a Banach space of summabilityexponent p ≥ 2), and this solution is Hölder continuous inboth time and space.

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

This paper gives an approximation of the solution of the Boltzmann equation by stochastic interacting particle systems in a case of cut-off collision operator and small initial data. In this case,following the ideas of Mischler and Perthame, we prove the existence and uniqueness of the solution of this equation and also the existence and uniqueness of the solution of the associated nonlinear martingale problem. 
Then, we first delocalize the interaction by considering a mollified Boltzmann equation in which the interaction is averaged on cells of fixed size which cover the space. In this situation, Grahamand Méléard have obtained an approximation of the mollified solution by some stochastic interacting particle systems. Then we consider systems in which the size of the cells depends on the size of the system. We show that the associated empirical measures converge in law to a deterministic probability measure whose density flow is the solution of the full Boltzmann equation. That suggests an algorithm based on the Poisson interpretation of the integral term for the simulation of this solution.

We study the evolution of a multi-component system which is modeled by a semi-Markov process. We give formulas for the avaibility and the reliability of the system. In the r-positive case, we prove that the quasi-stationary probability on the working states is the normalised left eigenvector of some computable matrix and that the asymptotic failure rate is equal to the absolute value of the convergence parameter r.

The growth exponent α for loop-erased or Laplacian random walkon the integer lattice is defined by saying that the expected time toreach the sphere of radius n is of order nα . We prove thatin two dimensions, the growth exponent is strictly greater than one.The proof uses a known estimate on the third moment of the escapeprobability and an improvement on the discrete Beurling projection theorem.

We establish posterior consistency for non-parametric Bayesian estimation of thedispersion coefficient of a time-inhomogeneous Brownian motion.

We prove a new large deviation inequality with applicationswhen projecting a density on a wavelet basis.