Consider a proper metric space and a sequence ${\left(Fₙ\right)}_{n\ge 0}$ of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) $X{ₙ}^x=Fₙ\circ ...\circ F₁\left(x\right)$ starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic...

We present a probabilistic model of the microscopic scenario of dielectric relaxation. We prove a limit theorem for random sums of a special type that appear in the model. By means of the theorem, we show that the presented approach to relaxation phenomena leads to the well known Havriliak-Negami empirical dielectric response provided the physical quantities in the relaxation scheme have heavy-tailed distributions. The mathematical model, presented here in the context of dielectric relaxation, can...

We consider a random walk in a random potential, which models a situation of a random polymer and we study the annealed and quenched costs to perform long crossings from a point to a hyperplane. These costs are measured by the so called Lyapounov norms. We identify situations where the point-to-hyperplane annealed and quenched Lyapounov norms are different. We also prove that in these cases the polymer path exhibits localization.

The aim of the paper is to establish strong laws of large numbers for sequences of blockwise and pairwise $m$-dependent random variables in a convex combination space with or without compactly uniformly integrable condition. Some of our results are even new in the case of real random variables.

Si considera, sul gruppo degli interi, una passeggiata aleatoria uscente dall’origine, i cui passi ammettano due soli possibili valori: uno strettamente negativo, l’altro strettamente positivo. Nel caso particolare in cui il primo di questi valori sia $-1$, si dà un’espressione esplicita per la legge del primo istante di ritorno nell’origine.

Soit $T_{\alpha }$ la rotation sur le cercle d’angle irrationnel $\alpha$, soit ${\left(S_k\right)}_{k\ge 0}$ une marche aléatoire transiente sur $\mathbb{Z}$. Soit $f\in L^2\left(\mu \right)$ et $H\in ]0,1[$, nous étudions la convergence faible de la suite $\frac{1}{n^H}{\sum }_{k=0}^{\left[nt\right]-1}f\circ T_{\alpha }^{S_k},n\ge 1.$

Let [0;a₁(x),a₂(x),…] be the regular continued fraction expansion of an irrational x ∈ [0,1]. We prove mainly that, for α > 0, β ≥ 0 and for almost all x ∈ [0,1], $li{m}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=\alpha /log2$ if α < 1 and β ≥ 0, $li{m}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=1/log2$ if α = 1 and β < 1, and, if α > 1 or α = 1 and β >1, $lim in{f}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=1/log2$, $lim su{p}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=\infty$, where $a{ⁿ}_i\left(x\right)=a_i\left(x\right)$ if $a_i\left(x\right)\le n^{\alpha }lo{g}^{\beta }n$ and $a{ⁿ}_i\left(x\right)=0$ otherwise, for all i ∈ 1,…,n.