Random walks in random scenery are processes defined by Zn:=∑k=1nωSk where S:=(Sk,k≥0) is a random walk evolving in Zd and ω:=(ωx,x∈Zd) is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk S and the random scenery ω, almost surely with respect to...

We introduce a broad class of self-similar processes {Z(t),t≥0} called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g,...

We present limit theorems for an asymmetric telegraph process with drift and jumps under different rescaling conditions. The explicit formulae for the related characteristic functions are derived by solving a Cauchy problem for the respective hyperbolic system.

The reduction of the number of parameters in high-order Markov chain already inspired several articles. In particular, Raftery proposed an autoregressive modelling which utilizes the same transition matrix, with a coefficient, for every lag. In this paper, we show that a model of the same type,...

A generalization of the Renormalization Group, which describes order-parameter fluctuations in finite systems, is developed in the specific context of percolation. This “Stochastic Renormalization Group” (SRG) expresses statistical self-similarity through a non-stationary branching process....

Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic Px(τBn)∼2/πσ−1VB(x)n−1/2 and provide an explicit formula for the limit VB as a function of...