Related papers: Kac-Stroock type approximations for the Brownian m…

We describe generalized Brownian motion related to parabolic equation systems from a logical point of view, i.e., as a generalization of Anderson's random walk. The connection to classical spaces is based on the Loeb measure. It seems that…

Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition…

We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution…

We consider a two parameter family of unitarily invariant diffusion processes on the general linear group $\mathbb{GL}_N$ of $N\times N$ invertible matrices, that includes the standard Brownian motion as well as the usual unitary Brownian…

A classical random walk $(S_t, t\in\mathbb{N})$ is defined by $S_t:=\displaystyle\sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in\mathbb{N}}$ are a one-order Markov chain, a short memory is introduced in the…

An $\al$-permanental process $\{X_{ t},t\in T \}$ is a stochastic process determined by a kernel $K=\{K(s,t),s,t\in T \}$, with the property that for all $t_{1},\ldots,t_{n}\in T $, $ |I+K( t_{1},\ldots,t_{n}) S|^{- \al} $ is the Laplace…

For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$.…

We consider a super-Brownian motion $\{X_t, t\geq 0\}$ in a random environment described by a centered Gaussian field $\{W(t,x),t\geq 0, x\in\mathbb{R}^d\}$ whose correlation function is given by $\mathcal{C} (x,y)(t \wedge s)$. The process…

We present a new technique for proving empirical process invariance principle for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound…

Let $L_{n}$ be the least common multiple of a random set of integers obtained from $\{1,\ldots,n\}$ by retaining each element with probability $\theta\in (0,1)$ independently of the others. We prove that the process $(\log L_{\lfloor…

Consider non-intersecting Brownian motions on the real line, starting from the origin at t=0, with a number of particles forced to reach p distinct target points at time t=1. This work shows that the transition probability, that is the…

The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed…

In this paper we provide sufficient conditions for sequences of stochastic processes of the form $\int_{[0,t]} f_n(u) \theta_n(u) du$, to weakly converge, in the space of continuous functions over a closed interval, to integrals with…

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an $\alpha$-stable law…

Let $X=(X_t)_{t\geq 0}$ be a known process and $T$ an unknown random time independent of $X$. Our goal is to derive the distribution of $T$ based on an iid sample of $X_T$. Belomestny and Schoenmakers (2015) propose a solution based the…

Given $n$ equidistant realisations of a L\'evy process $(L_t,\,t\ge 0)$, a natural estimator $\hat N_n$ for the distribution function $N$ of the L\'evy measure is constructed. Under a polynomial decay restriction on the characteristic…

The circular Dyson Brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain parameter-dependent ensembles of unitary random matrices. This model is considered with the…

In this paper we introduce a general stochastic representation for an important class of processes with resetting. It allows to describe any stochastic process intermittently terminated and restarted from a predefined random or non-random…

Consider a sequence {X(i,0) : i = 1, ..., n} of i.i.d. random variables. Associate to each X(i,0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy. In this way, we obtain i.i.d.…

We obtain complementary recurrence and transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi_{n+1}$. Here $M$ denotes a primitive matrix having…