Uniform convergence of basic Fourier–Bessel series on a q-linear grid

We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.

     

Convergence in Measure and τ-Compactness of τ-Measurable Operators,Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

     

Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law

We consider a continuous-time symmetric branching random walk on the d-dimensional lattice, d ≥?1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a continuous-time Markov branching process (a continuous-time analog of a Bienamye-Galton-Watson process) at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We replay why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d =?1 and d =?2.

     

Vague Convergence of Semimartingale Random Measures

Abstract

In this paper, we introduce and research the vague convergence of semimartingale random measures in distribution. The conditions are provided for the vague convergence of semimartingale random measures and the convergence of stochastic integrals with respect to semimartingale random measures in distribution.     

Stationary solutions of random differential equations with polynomial nonlinearities

The paper deals with systems of ODEs containing polynomial nonlinearities and random inhomogeneous terms. Applying perturbation method pathwise solutions are found in form of power series with respect to a parameter η controlling the nonlinearities. Under the assumption that for η=0 the system is stable and that the inhomogeneous terms are bounded the radius of convergence of the perturbation series is estimated. Further, it is proved that the perturbation series form stationary solutions if the inhomogeneous terms are stationary.     

On Complete Convergence for Arrays of Random Elements and Variables

Abstract

We obtain complete convergence results for arrays of row-wise independent Banach space valued random elements. The main result deals with two cases that usually are considered separately: when no assumptions are made concerning the geometry of the underlying Banach space and when the Banach space is of Rademacher type p.     

On Complete Convergence for Arrays of Random Elements

Abstract

A complete convergence theorem for arrays of rowwise independent random variables was obtained by Kruglov, Volodin, and Hu (Statistics and Probability Letters 2006, 76:1631–1640). In this article, we extend the result to a Banach space without any additional conditions. No assumptions are made concerning the geometry of the underlying Banach space.     

Uniform convergence of fuzzy random renewal process

Recently, Zhao et al. (in Fuzzy Optimization and Decision Making 2007 6, 279–295) presented a fuzzy random elementary renewal theorem and fuzzy random renewal reward theorem in the fuzzy random process. In this paper, we study the convergence of fuzzy random renewal variable and of the total rewards earned by time t with respect to the extended Hausdorff metrics d _∞ and d ₁. Using this convergence information and applying the uniform convergence theorem, we provide some new versions of the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem.     

Series of independent random variables in rearrangement invariant spaces: An operator approach

This paper studies series of independent random variables in rearrangement invariant spacesX on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(L _p ), 1≤p≤∞ and Lorentz spacesA _Ψ. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumptionL _p ⊂X, p<∞ is sufficient to guarantee that convergence of such series inX (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence inX of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result. Research supported by the Australian Research Council.     

Strong Law of Large Numbers for Stationary Sequences of Random Upper Semicontinuous Functions

Abstract

The strong laws of large numbers with the convergence in the sense of the uniform Hausdorff metric for stationary sequences of random upper semicontinuous functions is established. This approach allows us to deduce many results on the convergence in uniform Hausdorff metric of random upper semicontinuous functions from the relevant results on real-valued random variables that appear as their support functions.     

Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopf ^d , in the geometric decay case, rates have the form n ^−1/(8d+24) L(n), where L(n) is a power of log(n). This revised version was published online in June 2006 with corrections to the Cover Date.     

A borderline Gaussian random Fourier series for the sample convergence in variation

We give an example of a Gaussian random Fourier series, of which the normalized remainders have their sojourn times converging in variation, namely the convergence in the space L¹(R) of the related density distributions, to the Gaussian density. The convergence in the space L^∞(R) is also proved. In our approach, we use local times of Gaussian random Fourier series.     

行m-NSD随机变量阵列的完全收敛性

本文研究了行m-NSD随机变量阵列的完全收敛性问题.主要利用m-NSD随机变量的Kolmogorov型指数不等式,获得了行m-NSD随机变量阵列的完全收敛性定理,将Hu等（1998）andSung等（2005）的结果从独立情形推广到了m-NSD随机变量阵列.本文的结论同样推广了Chen等（2008）,Hu等（2009）,Qiu等（2011）和Wang等（2014）的结果.     

Convergence rates of the blocked Gibbs sampler with random scan in the Wasserstein metric

ABSTRACT

This paper establishes explicit estimates of convergence rates for the blocked Gibbs sampler with random scan under the Dobrushin conditions. The estimates of convergence in the Wasserstein metric are obtained by taking purely analytic approaches.     

Bibounded uo-convergence and b-property in vector lattices

We define bidual bounded uo-convergence in vector lattices and investigate relations between this convergence and b-property. We prove that for a regular Riesz dual system \(\langle X,X^{\sim }\rangle \), X has b-property if and only if the order convergence in X agrees with the order convergence in \(X^{\sim \sim }\).

     

Berry-Esseen type estimations for multi-sample U-statistics

We study the rate of convergence in the central limit theorem for nondegenerate multi-sample U-statistics of a series of independent samples of independent random variables under minimal sufficient moment conditions on the canonical functions of the Hoeffding representation. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 69–90.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript> convergence for <Emphasis Type="Italic">B</Emphasis>-valued random elements

The paper investigates L ^p convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 < p < 2. The paper also discusses L ^r convergence and L ^r bound for random elements without any geometric restriction condition on the Banach space.     

Using Random Quasi-Monte-Carlo Within Particle Filters,With Application to Financial Time Series

This article presents a new particle filter algorithm which uses random quasi-Monte-Carlo to propagate particles. The filter can be used generally, but here it is shown that for one-dimensional state-space models, if the number of particles is N, then the rate of convergence of this algorithm is N^?1. This compares favorably with the N^?1/2 convergence rate of standard particle filters. The computational complexity of the new filter is quadratic in the number of particles, as opposed to the linear computational complexity of standard methods. I demonstrate the new filter on two important financial time series models, an ARCH model and a stochastic volatility model. Simulation studies show that for fixed CPU time, the new filter can be orders of magnitude more accurate than existing particle filters. The new filter is particularly efficient at estimating smooth functions of the states, where empirical rates of convergence are N^?3/2; and for performing smoothing, where both the new and existing filters have the same computational complexity.     

Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications

In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on L _p convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.     

Almost sure convergence of smoothingD m -splines for noisy data

Summary The purpose of this paper is to study the convergence of smoothingD ^m -splines relative to sets of data perturbed by a random noise. Conditions of almost sure convergence and error estimates are given.