We prove that the motion of a test particle in a hard sphere fluid in thermal equilibrium converges, in the Boltzmann-Grad limit, to the stochastic process governed by the linear Boltzmann equation. The convergence is in the sense of weak convergence of the path measures. We use this result to study the steady state of a binary mixture of hard spheres of different colors (but equal masses and d...

A common assumption in the modeling of stochastic processes is that of weak stationarity. Although this is a convenient and sometimes justifiable assumption for many applications, there are other applications for which it is clearly inappropriate. One such application occurs when the process is driven by action at a limited number of sites, or point sources. Interest may lie not only in predict...

We consider a particle in the over-damped regime at zero temperature under the influence of a sawtooth potential and of a noisy force, which is correlated in time. A current occurs, even if the mean of the noisy force vanishes. We calculate the stationary probability distribution and the stationary current. We discuss, how these items depend on the characteristic parameters of the underlying st...

The question whether a time series behaves as a random walk or as a stationary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This paper studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptoti...

Brownian motion is a stochastic process (Wt, t ≥ 0) such that • W0 = 0. • (Wt2 −Wt1) and (Wt4 −Wt3) are independent, for any t1 < t2 ≤ t3 < t4. • (Wt2 −Wt1) ∼ N(0, t2 − t1). • For almost all ω, the sample function t 7→Wt(ω) is continuous. Since Wt has stationary independent increments, we already know that EWt = 0 and that the covariance function is r(t, s) = t∧ s. Thus the covariance matrix of...

By using multiple Wiener-Itô stochastic integrals, we study the cubic variation of a class of selfsimilar stochastic processes with stationary increments (the Rosenblatt process with selfsimilarity order H ∈ ( 12 , 1)). This study is motivated by statistical purposes. We prove that this renormalized cubic variation satis es a non-central limit theorem and its limit is (in the L(Ω) sense) still ...

We survey distributional properties of R-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in R) which determine recurrence or transience. Let (Xn, n ≥ 0) be an ergodic stationary Rd-valued stochastic process, and let (Yn = X0 + · · · + Xn−1, n ≥ 1) be the associated random walk. What can one say about recurrence of this random wa...

There is wide interest in weakly nonlinear wave equations with random initial data. A common approach is the approximation through a kinetic transport equation, which clearly poses the issue of understanding its validity in the kinetic limit. While for the general case a proof of the kinetic limit remains open, we report here on first progress. As wave equation we consider the nonlinear Schrödi...

Probability distributions are central tools for probabilistic modeling in data mining. In functional data analysis (FDA) they are weakly studied in the general case. In this paper we discuss a probability distribution law for functional data considered as stochastic process. We define first a new kind of stationarity linked t o the Archimedean copulas, and then we build a probability distributi...

We study an infinite-server queue with a general arrival process and a large class of general time-varying service time distributions. Specifically, customers’ service times are conditionally independent given their arrival times, and each customer’s service time, conditional on her arrival time, has a general distribution function. We prove functional limit theorems for the two-parameter proce...