Let ${\left(M_n\right)}_{n\ge 0}$ be a zero-mean martingale with canonical filtration ${\left({ℱ}_n\right)}_{n\ge 0}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,...,$ which means that $P\left(\right|Y_n|>t|{ℱ}_{n-1})\le 1-H\left(t\right)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2={\sum }_{n\ge 1}E\left(Y_n^2\right|{ℱ}_{n-1})$. It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P(M_n\in [-c,c]i.o.|V=\infty )=1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions....

We consider random walks where each path is equipped with a random weight which is stationary and independent in space and time. We show that under some assumptions the arising probability distributions are in a sense uniformly absolutely continuous with respect to the usual probability distribution for symmetric random walks.

We show that the norm of the random walk operator on the Cayley graph of the surface group in the standard presentation is bounded by 1/√g where g is the genus of the surface.

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1-\Phi \left(z\right)$ inside the unit disk, where $\Phi$ is the generating function of $X$, under some mild conditions

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $logt$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar...

We prove an extension of a result by Peres and Solomyak on almost sure absolute continuity in a class of symmetric Bernoulli convolutions.

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.