An Extension of the Kolmogorov–Feller Weak Law of Large Numbers with an Application to the St. Petersburg Game

The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables without finite mean is extended to a larger class of distributions, requiring regularly varying normalizing sequences. As an application we show that the weak law of large numbers for the St. Petersburg game is an immediate consequence of our result.     

弱Ⅰ序列的刻画

在文献[5]中,周才军定义了弱Ⅰ序列,并利用Koszul上同调和局部上同调的方法刻画了这种序列.本文利用Ext函子刻画了弱Ⅰ序列.     

Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

In this paper, we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong and weak convergence of the sequences.     

■混合阵列行加权和最大值的若干收敛性质

讨论了■混合阵列行加权和最大值的弱收敛性、L_p收敛性和完全收敛性定理,将独立阵列的相关极限定理推广到了■混合随机阵列情形.     

软弹簧系统在周期小扰动下通向浑沌的道路

本文用Mel'nikov-Holmes方法讨论了软弹簧系统在周期小扰动作用下进入浑沌的途径.发现对于这类系统,由于所加扰动的方式不同,系统会以不同次谐分叉序列通向浑沌,且随着外加频率的变化这种序列会以不同的排列方式把系统导入浑沌.     

Degeneration and regularization of the operator in the Cauchy problem for the Schrödinger equation

The Cauchy problem for the Schrödinger equation whose operator degenerates on a half-line is studied. In order to approximate a solution to the problem with degeneracy by solutions to well-posed problems, the notion of regularization for an operator with degeneracy is introduced; an approximative solution to a problem with degeneracy is defined as the limit of a sequence of regularized problems. The dependence of the approximative solution on the choice of the class of admissible regularizations is studied. The weak compactness of sequences of states determined by sequences of solutions to regularized problems in the topologies determined by the space of all bounded linear operators and by subspaces of mutually commuting bounded linear operators is investigated.     

An Extension of the Weak Law of Large Numbers for Exchangeable Sequences

We prove an extension of the Kolmogorov-Feller weak law of large numbers for exchangeable sequences, by relaxing the finite mean condition therein.     

弱$I$序列的刻画

在文献[5]中,周才军定义了弱I序列,并利用Koszul上同调和局部上同调的方法刻画了这种序列,本文利用Ext函子刻画了弱I序列。     

Asymptotics for statistical functionals of long-memory sequences

We present two general results that can be used to obtain asymptotic properties for statistical functionals based on linear long-memory sequences. As examples for the first one we consider L- and V-statistics, in particular tail-dependent L-statistics as well as V-statistics with unbounded kernels. As an example for the second result we consider degenerate V-statistics. To prove these results we also establish a weak convergence result for empirical processes of linear long-memory sequences, which improves earlier ones.     

Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems

In this paper, we introduce some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of nonspreading mappings and a finite family of nonexpansive multivalued mappings in Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. The results obtained in this paper extend and improve some recent known results.     

Lipschitz Functions and Ekeland’s Theorem

As shown by F. Sullivan (Proc. Am. Math. Soc. 83:345–346, 1981), validity of the weak Ekeland variational principle implies completeness of the underlying metric space. In this note, we show that what really forces completeness in Sullivan’s argument is an even simpler geometric property of lower bounded Lipschitz functions. We derive the weak Ekeland principle from this new property, and use the new property to directly obtain an omnibus non-empty intersection result for decreasing sequences of closed sets that yields as special cases the theorems of Cantor and Kuratowski valid in complete metric spaces     

Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

This paper is concerned with large-$\text{O}$ error estimates concerning convergence in distribution as well as norm convergence for Banach space-valued martingale difference sequences. Indeed, two general limit theorems equipped with rates of convergence for such difference sequences are established. Applications of these lead to the central limit theorem and the weak law of large numbers with rates for Banach space-valued martingales.     

Holderian Weak Invariance Principle for Stationary Mixing Sequences

We provide some sufficient mixing conditions on a strictly stationary sequence in order to guarantee the weak invariance principle in Hölder spaces. Strong mixing and \(\rho \)-mixing conditions are investigated as well as \(\tau \)-dependent sequences. The main tools are deviation inequalities for mixing sequences.     

关于两两NQD列的若干极限性质

本文讨论了两两ＮＱＤ列的Ｍａｒｃｉｎｋｉｅｗｉｃｚ型弱大数律及Ｊａｍｉｓｏｎ型加权和的强稳定性，指出了两两ＮＱＤ列与独立列的一个本质性区别，从而提出了解决两两ＮＱＤ旬几乎处处收敛和完全收敛的一种方式。     

Weak solutions to stochastic 3D Navier-Stokes-α model of turbulence: α-Asymptotic behavior

We study the asymptotic behavior of weak solutions to the stochastic 3D Navier-Stokes-α model as α approaches zero. The main result provides a new construction of the weak solutions of stochastic 3D Navier-Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier-Stokes-α model.     

Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions

We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at the moment. In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one.     

Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process     

非平稳φ-混合序列强律收敛速度的进一步探讨

本文研究非平稳φ 混合序列的强大数律：无需混合速度限制，给出了完全收敛成立的充要条件；在弱混合速度条件下，讨论了Ｍａｒｃｉｎｋｉｅｗｉｃｚ Ｚｙｇｍｕｎｄ强律成立的必要条件，进而得出混合情形的Ｍａｒｃｉｎｋｉｅｗｉｃｚ Ｚｙｇｍｕｎｄ强律，其结果与独立情形相一致．     

Structural properties of Markov chains with weak and strong interactions

Markov chains have been frequently used to characterize uncertainty in many real-world problems. Quite often, these Markov chains can be decomposed into a vector consisting of fast and slow components; these components are coupled through weak and strong interactions. The main goal of this work is to study the structural properties of such Markov chains. Under mild conditions, it is proved that the underlying Markov chain can be approximated in the weak topology of L² by an aggregated process. Moreover, the aggregated process is shown to converge in distribution to a Markov chain as the rate of fast transitions tends to infinity. Under an additional Lipschitz condition, error bounds of the approximation sequences are obtained.     

Weak and Strong Density of Compositions

The convergence in various topologies of sequences of inner superposition (composition) operators acting between Lebesgue spaces and of their linear combinations is studied. In particular, the sequential density results for the linear span of such operators is proved for the weak, weak continuous and strong operator topologies.