The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.     

A perturbed ergodic theorem

Using a version of an ergodic lemma due to Cuculescu and Foias, we prove a pointwise ergodic theorem for -contractions which can be viewed as a perturbed version of the celebrated ergodic theorem of Chacon and Ornstein. Surprisingly, to some extent, the complex part of the iterates involved have no effect on the ergodic convergence.

     

Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

A mean ergodic theorem on vector spaces

A mean ergodic theorem for a matrix is first proved from which a mean ergodic theorem for affine operators on a vector space without any topological structure is obtained.     

A non-singular dynamical system without maximal ergodic inequality

It is shown that there is a non-singular dynamical system for which the maximal ergodic inequality does not hold. We further discuss the connection between a non-singular dynamical systems and the pointwise convergence of the Furstenberg ergodic averages.     

A note on the almost sure limit theorem for U-statistic

Summary In this note we prove an almost sure limit theorem for the products of U-statistics.     

Applications of the ergodic iteration theorem

I prove several natural preservation theorems for the countable support iteration. This solves a question of Ros?anowski regarding the preservation of localization properties and greatly simplifies the proofs in the area (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tutte's spring theorem

We present a short proof of the theorem of Tutte that every planar 3‐connected graph has a drawing in the plane such that every vertex which is not on the outer cycle is the barycenter of its neighbors. Moreover, this holds for any prescribed representation of the outer cycle. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 275–280, 2004     

A Harris‐Kesten theorem for confetti percolation

Percolation properties of the dead leaves model, also known as confetti percolation, are considered. More precisely, we prove that the critical probability for confetti percolation with square‐shaped leaves is 1/2. This result is related to a question of Benjamini and Schramm concerning disk‐shaped leaves and can be seen as a variant of the Harris‐Kesten theorem for bond percolation. The proof is based on techniques developed by Bollobás and Riordan to determine the critical probability for Voronoi and Johnson‐Mehl percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 361–385, 2015     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

Individual ergodic theorem for unitary maps of random matrices

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices.

     

Tychonoff's theorem for locally compact spaces and an elementary approach to the topology of path spaces

The path spaces of a directed graph play an important role in the study of graph -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.

     

Sequential entry and exit decisions with an ergodic performance criterion

We consider a variant of an optimisation problem involving sequential entry and exit decisions that has emerged in the economics literature as a real option model. The problem that we solve aims at maximising an ergodic, or long-term average, performance criterion in a pathwise as well as in an expected sense. Such a performance index is probably better suited to decision making within a sustainable economic environment. Our results include a complete characterisation of the optimal strategy, which can take qualitatively different forms depending on the problem's data, as well as explicit expressions for the maximal value of the associated performance index.