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.

     

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)     

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.

     

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.     

Little's theorem: A stochastic integral approach

In a previous paper we have given a unified approach to the PASTA and the conditional PASTA property that is based upon the observation that the difference between the two limits can be represented as a stochastic integral with respect to a square integrable martingale. The equality of the two limits is then a consequence of a strong law of large numbers for martingales. In this paper we derive a non-standard version of Little's theorem via the same method. The moral of the story is that each of these theorems is but a particular case of a more general theory.     

Multiparameter ratio ergodic theorems for semigroups

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chacon?s type continuous ratio ergodic theorems for semigroups and of Jacobs? type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Sucheston?s type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chacon?s local ratio ergodic theorem for semigroups of positive linear contractions on _L1$L_1$ is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.     

A local convergence theorem for partial sums of stochastic adapted sequences

In this paper we establish a new local convergence theorem for partial sums of arbitrary stochastic adapted sequences. As corollaries, we generalize some recently obtained results and prove a limit theorem for the entropy density of an arbitrary information source, which is an extension of case of nonhomogeneous Markov chains.     

一类随机适应序列部分和的局部极限定理的一个注记

研究了一类随机适应序列的强极限定理,推广了最近发表的几个结果,并进一步推广了Borel—Cantelli引理．     

Subadditive ergodic theorems for random sets in infinite dimensions

We prove pointwise and mean versions of the subadditive ergodic theorem for superstationary families of compact, convex random subsets of a real Banach space, extending previously known results that were obtained in finite dimensions or with additional hypotheses on the random sets. We also show how the techniques can be used to obtain the strong law of large numbers for pairwise independent random sets, as well as results in the weak topology.     

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)$p>d/(d-1)$.     

关于有信用风险的期权定价的鞅方法

本文考虑了一个关于具有对方风险的衍生物的金融模型\bd 应用公司价值模型, 本文讨论了关于具有对方破产风险的衍生物的欧式期权定价问题\bd 应用鞅方法, 在高斯分布等的假设下本文得到并证明一个关于该期权的显式Black-Scholes定价公式\bd 该公式推广了Ammann在[1]中的相应结果.     

Semi-uniform sub-additive ergodic theorems for skew-product quasi-flows

The aim of this paper is to extend the semi-uniform ergodic theorem and semi-uniform sub-additive ergodic theorem to skew-product quasi-flows. Furthermore, more strict inequalities about these two theorems are established. By making use of these results, it is feasible to get uniform estimation of the Lyapunov exponent of some special systems even under non-uniform hypotheses     

An abstract ergodic theorem and some inequalities for operators on Banach spaces

We prove an abstract mean ergodic theorem and use it to show that if is a sequence of commuting -dissipative (or normal) operators on a Banach space , then the intersection of their null spaces is orthogonal to the linear span of their ranges. It is also proved that the inequality holds for any -dissipative operator . These results either generalize or improve the corresponding results of Shaw, Mattila, and Crabb and Sinclair, respectively.

     

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 central limit theorem for the sojourn times of strongly ergodic Markov chains

Let X_n be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if $\text{t}{}^{\mathrm{(n}}\text{)}{}_{\mathrm{i}}\text{=Σ}{}^{\mathrm{n}}{}_{\mathrm{k=1}}\text{i}\text{?}{}_{\mathrm{i}}\text{(X}{}_{\mathrm{k}}\text{)}$, then the probability measures induced by {t⁽ⁿ⁾_i/√n?√nπ_i}_i?I(π_i being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator.     

Imprecise stochastic processes in discrete time: global models,imprecise Markov chains,and ergodic theorems

We justify and discuss expressions for joint lower and upper expectations in imprecise probability trees, in terms of the sub- and supermartingales that can be associated with such trees. These imprecise probability trees can be seen as discrete-time stochastic processes with finite state sets and transition probabilities that are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures. We derive various properties for their joint lower and upper expectations, and in particular a law of iterated expectations. We then focus on the special case of imprecise Markov chains, investigate their Markov and stationarity properties, and use these, by way of an example, to derive a system of non-linear equations for lower and upper expected transition and return times. Most importantly, we prove a game-theoretic version of the strong law of large numbers for submartingale differences in imprecise probability trees, and use this to derive point-wise ergodic theorems for imprecise Markov chains.     

A Poisson limit theorem for a strongly ergodic non-homogeneous Markov chain

A strongly ergodic non-homogeneous Markov chain is considered in the paper. As an analog of the Poisson limit theorem for a homogeneous Markov chain recurring to small cylindrical sets, a Poisson limit theorem is given for the non-homogeneous Markov chain. Meanwhile, some interesting results about approximation independence and probabilities of small cylindrical sets are given.