Related papers: Pointwise Convergence for Random Ergodic Averages …

In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem.…

The paper is primarily concerned with the asymptotic behavior as $N\to\infty$ of averages of nonconventional arrays having the form $N^{-1}\sum_{n=1}^N\prod_{j=1}^\ell T^{P_j(n,N)}f_j$ where $f_j$'s are bounded measurable functions, $T$ is…

Let ${\bf X}=(X, \Sigma, m, \tau)$ be a dynamical system. We prove that the bilinear series $\sideset{}{'}\sum_{n=-N}^{N}\frac{f(\tau^nx)g(\tau^{-n}x)}{n}$ converges almost everywhere for each $f,g\in L^{\infty}(X).$ We also give a proof…

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages, $\Sigma_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic trajectory $(x_n)$ of a function $T$ converges to the space average $\int f d\mu$, where…

We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages…

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our…

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge…

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

Let $\Gamma$ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action $\Gamma \curvearrowright (X, \mu)$ and a map $f \in L^1(X, \mu)$, and to compare the global…

This paper is devoted to the study of various maximal ergodic theorems in noncommutative $L_p$-spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive…

We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n)…

In this article, we extend the Bufetov pointwise ergodic theorem for spherical averages of even radius for free group actions on noncommutative $L\log L$-space. Indeed, we extend it to more general Orlicz space $L^\Phi(M, \tau)$…

Let $U$ be a unitary operator acting on the Hilbert space H, and $\alpha:\{1,..., m\}\mapsto\{1,..., k\}$ a partition of the set $\{1,..., m\}$. We show that the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1}…

In two recent papers we introduced some new techniques for constructing an extension of a probability-preserving system $T:\mathbb{Z}^d\curvearrowright (X,\mu)$ that enjoys certain desirable properties in connexion with the asymptotic…

Local mean and individual (with respect to almost uniform convergence in Egorov's sense) ergodic theorems are established for actions of the semigroup $\mathbb R_+^d$ in symmetric spaces of measurable operators associated with a semifinite…

We study the rate of growth of ergodic sums along a sequence (a_n) of times: S_N f(x)=f(T^{a_1}x) + ... + f(T^{a_N}x). We characterize the maximal rate of growth of these ergodic sums and identify a number of sequences such as (2^n) that…

Let $U$ be a unitary operator acting on the Hilbert space $H$, $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair--partition, and finally $A_{1},...,A_{2k-1}\in B(H)$. We show that the ergodic average $$…

N. Fournier and A. Guillin obtained in their 2015 PTRF paper some bounds of the L^p-mean rate of convergence in Wasserstein distance of empirical distributions for a class of stationary mixing processes. In this paper, we propose to extend…

Let $(\xi_n)_{n=0}^\infty$ be a nonhomogeneous Markov chain taking values from finite state-space of $\mathbf{X}=\{1,2,\ldots,b\}$. In this paper, we will study the generalized entropy ergodic theorem with almost-everywhere and…

We prove the $L^{2}$ convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, ..., T_{l}$, assuming that each map $T_i$ and each pair $T_iT_j^{-1}$ is ergodic for $i\neq j$. The limiting behavior of such…