We prove that the Poisson distribution maximises entropy in the class of ultra log-concave distributions, extending a result of Harremoës. The proof uses ideas concerning log-concavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is...

We consider Sinai's random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is bounded almost surely. We also get that i.o. the maximal distance between two favorite sites is bounded almost surely.

Let X1,X2,... be independent variables, each having a normal distribution with negative mean -[beta]<0 and variance 1. We consider the partial sums Sn=X1+...+Xn, with S0=0, and refer to the process {Sn:n>=0} as the Gaussian random walk. This paper is concerned with the cumulants of the maximum M[beta]=max{Sn:n=0}. We express all cumulants of M[beta] in terms of Taylor series about [beta] at 0 with...</0>

In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one- dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This...

Given a random time, we give some characterizations of the set of martingales for which the stopping theorems still hold. We also investigate how the stopping theorems are modified when we consider arbitrary random times. To this end, we introduce some families of martingales with remarkable...

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order...

A tempered stable Lévy process combines both the [alpha]-stable and Gaussian trends. In a short time frame it is close to an [alpha]-stable process while in a long time frame it approximates a Brownian motion. In this paper we consider a general and robust class of multivariate tempered stable...

We prove the convergence and the asymptotic normality of the quadratic variations of the spherical fractional Brownian motion.

We prove existence and (in some special case) uniqueness of an invariant measure for the transition semigroup associated with the stochastic wave equations with nonlinear dissipative damping.

Theorems of approximation of Gaussian processes for the sequential empirical process of the permutations of independent random variables are established. The results are applied to simulate critical values for the functionals of sequential empirical processes used in change point analysis. The...