Related papers: Ergodic averages along sequences of slow growth

The idea of a parsing of a stationary process according to a collection of words is introduced, and the basic framework required for the asymptotic analysis of these parsings is presented. We demonstrate how the pointwise ergodic theorem…

In the first part of the paper the natural scheme for proving noncommutative individual ergodic theorems for multiple sequences is described and applied to obtain results on unrestricted convergence of multiaverages. In the second part…

We examine the limiting behavior of multiple ergodic averages associated with arithmetic progressions whose differences are elements of a fixed integer sequence. For each $\ell$, we give necessary and sufficient conditions under which…

This paper considers a class of nonautonomous slow-fast stochastic partial differential equations driven by $\alpha$-stable processes for $\alpha\in (1,2)$. By introducing the evolution system of measures, we establish an averaging…

For a continuous semicascade on a metrizable compact set $\Omega $, we consider the weak$^{*}$ convergence of generalized operator ergodic means in ${\rm End}\, \, C^{*} (\Omega)$. We discuss conditions on the dynamical system under which…

We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation…

Szemer\'edi's Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain…

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.

We show an averaging result for a system of stochastic evolution equations of parabolic type with slow and fast time scales. We derive explicit bounds for the approximation error with respect to the small parameter defining the fast time…

We establish a pointwise convergence result for ergodic averages modeled along orbits of the form $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$, where $k$ is an arbitrary positive rational number with $\sqrt{k}\not\in\mathbb{Q}$. Namely,…

We prove an Erd\H{o}s-Kac type of theorem for the set $S(x,y)=\{n\leq x: p|n \Rightarrow p\leq y \}$. If $\omega (n)$ is the number of prime factors of $n$, we prove that the distribution of $\omega(n)$ for $n \in S(x,y)$ is Gaussian for a…

We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are applied to homogenize convection dominated parabolic equations with rapidly…

We exhibit rationally ergodic, weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.

We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplica-tive ergodic theorem with an…

We study in this paper the validity of the mean ergodic theorem along \emph{left} F\o lner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left F\o lner sequence in $G$, we…

We develop a framework for the study of the limiting behavior of multiple ergodic averages with commuting transformations when all iterates are given by the same sparse sequence; this enables us to partially resolve several longstanding…

Let $\Omega(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N…

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space of measurable functions on $(\Omega,\mu)$. If $\mu(\Omega)=\infty$, necessary and sufficient conditions are…

We investigate three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/\Gamma$ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from…

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to…