Uniform Integrability Research Articles

Forward-convex convergence in probability of sequences of nonnegative random variables

For a sequence $ (f_n)_{n \in \mathbb{N}}$ of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence $ (h_n)_{n \in \mathbb{N}}$ with $ h_n\in \mathrm {conv}(\{f_n,f_{n+1},\dots \})$ for all $ n \in \mathbb{N}$ to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.

STRONG FELLER CONTINUITY OF FELLER PROCESSES AND SEMIGROUPS

We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.

Laws of large numbers for Cesàro alpha-integrable random variables under dependence condition AANA or AQSI

Both residual Cesaro alpha-integrability (RCI(α)) and strongly residual Cesaro alpha-integrability (SRCI(α)) are two special kinds of extensions to uniform integrability, and both asymptotically almost negative association (AANA) and asymptotically quadrant sub-independence (AQSI) are two special kinds of dependence structures. By relating the RCI(α) property as well as the SRCI(α) property with dependence condition AANA or AQSI, we formulate some tail-integrability conditions under which for appropriate α the RCI(α) property yields L 1-convergence results and the SRCI(α) property yields strong laws of large numbers, which is the continuation of the corresponding literature.

Poly(vinyl alcohol) reinforced with large-diameter carbon nanotubes via spray winding

For practical application of carbon nanotube (CNT)/polymer composites, it is critical to produce the composites at high speed and large scale. In this study, multi-walled carbon nanotubes (MWNTs) with large diameter (∼45nm) and polyvinyl alcohol (PVA) were used to increase the processing speed of a recently developed spraying winding technique. The effect of the different winding speed and sprayed solution concentration to the performance of the composite films were investigated. The CNT/PVA composites exhibit tensile strength of up to 1GPa, and modulus of up to 70GPa, with a CNT weight fraction of 53%. In addition, an electrical conductivity of 747S/cm was obtained for the CNT/PVA composites. The good mechanical and electrical properties are attributed to the uniform CNTs and PVA matrix integration and the high degree of tube alignment.

Convergence in r-mean of weighted sums of NQD random variables

Under some conditions of uniform integrability, the L r convergence for weighted sums of arrays of rowwise linearly negative quadrant dependent random variables has been established by Wu and Guan [Y. Wu, M. Guan, Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables, J. Math. Anal. Appl. 377 (2011) 613–623]. In this paper, we point out a gap in the proof and extend the result to rowwise pairwise negative quadrant dependent (NQD) random variables under weaker uniformly integrable conditions.

A Numerical Scheme for Invariant Distributions of Constrained Diffusions

Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady-state performance of the corresponding stochastic networks, and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions. In this work we propose and analyze a Monte-Carlo scheme based on an Euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity. Appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model. Almost sure consistency results are established That, in particular, show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a.s. to the corresponding integrals with respect to the invariant measure. Proofs rely on constructing suitable Lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of Echeverria's criteria for reflected diffusions. Regularity properties of the underlying Skorohod problems play a key role in the proofs. Rates of convergence for suitable families of test functions are also obtained. A key advantage of Monte-Carlo methods is the ease of implementation, particularly for high-dimensional problems. A numerical example of an eight-dimensional Skorohod problem is presented to illustrate the applicability of the approach.

Triangular set functions and the Nikodym, Brooks-Jewett, and Vitali-Hahn-Saks theorems on convergent sequences of measures

We consider triangular set functions which have no escaping load and are defined on classes of sets which are closed with respect to the union of at most countably many disjoint sets in the given class. We refine and generalize the Nikodym, Brooks-Jewett, and Vitali-Hahn-Saks theorems to these classes of set functions in a nontrivial manner; these include quasi-Lipschitz and finitely additive set functions, vector-valued measures, semimeasures, and outer measures. As immediate consequences of the theorems proved here we obtain a series of statements concerning the extension of properties of set functions, such as convergence, compactness, uniform absolute continuity and absolute continuity, from a ring of sets to the σ-ring generated by the ring.Bibliography: 18 titles.

Basic renewal theorems for random walks with widely dependent increments

In this paper, we derive some basic renewal theorems for random walks with widely dependent increments, which contain some common negatively dependent random variables (r.v.s.), some positively dependent r.v.s. and some others. For this purpose, we investigate uniform integrability for related counting processes and the strong law of large numbers for the widely dependent r.v.s.

Central limit theorem for positively associated stationary random fields

Positively associated stationary random fields on d-dimensional integral lattice arise in various models of mathematical statistics, percolation theory, statistical physics, and reliability theory. In this paper, we shall be concerned with a field with covariance functions satisfying a more general condition than summability. A criterion for the validity of the central limit theorem (CLT) for partial sums of a field from this class is established. The sums are taken over an increasing nest of parallelepipeds or cubes. The well-known conjecture of Newman stated that for an associated stationary random field the above condition on the covariance function should force the CLT to hold. As was shown by N. Herrndorf and A. P. Shashkin, this conjecture fails already for d = 1. In the present paper, the uniform integrability of the squared partial sums is shown as being of key importance for the CLT to hold. Thus, an extension of Lewis’s theorem proved for a sequence of random variables is obtained. Also, it is indicated how to modify Newman’s conjecture for any d. A representation of variances of partial sums of a field by means of slowly varying functions of several arguments is used in an essential way.

A note on moment convergence of bootstrap M-estimators

Abstract This paper studies the consistency of bootstrap moment estimators for a general M-estimator. We establish a theorem on the uniform integrability of the bootstrap M-estimator, thereby giving sufficient conditions for the consistency of the bootstrap moment estimators. As an application of our theorem, we provide sufficient conditions for the consistency of the bootstrap variance estimator for the quantile regression estimator, which has been considered as an important unsolved problem in the literature. We also discuss a justification of a bootstrap information criterion.

Almost sure central limit theorem for branching random walks in random environment

We consider the branching random walks in $d$-dimensional integer lattice with time--space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable. When $d\geq3$ and the fluctuation of environment satisfies a certain uniform square integrability then it is nondegenerate and we prove a central limit theorem for the density of the population in terms of almost sure convergence.

A local maximal inequality under uniform entropy

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant δ. The bound is expressed in the uniform entropy integral of the class at δ. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions.

Uniform Integrability for Strong Ratio Limit Theorems. II

We propose a new approach to obtaining strong limit theorems for ratio (SLTR). For this aim we use triplets, each of which consists of a transition operator for a Markov chain with a measurable state space $(E,\ccE)$, and also functions and measures given on E and $\ccE$, respectively. We say that a triplet is stable if it satisfies a somewhat strengthened statement of the simplest SLTR in the sense of part I of this paper [Theory Probab. Appl., 50 (2006), pp. 436–447]. It turns out, in particular, that under wide assumptions SLTR is reduced to testing the stability (sometimes in a weakened form) of various triplets. Incidentally we discuss SLTR for Markov chains having the truncated Liouville property.

Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

Under some conditions of uniform integrability and appropriate conditions, mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables are obtained. Our results extend and improve the results of [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008) 289–300] and [M. Ordóñez Cabrera, A. Volodin, Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305 (2005) 644–658].

A note on the de La Vallée Poussin criterion for uniform integrability

In this paper, we present new versions of the classical de La Vallée Poussin criterion for uniform integrability. Our results concern the uniform integrability of a continuous function relative to a sequence of distribution functions. We apply our results to obtain a result on the convergence of a sequence of integrals which we illustrate with an example.

Gibbs Random Fields with Unbounded Spins on Unbounded Degree Graphs

Gibbs fields are constructed and studied which correspond to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs of a certain type, for which the Gaussian Gibbs fields need not be existing. In these graphs, the vertex degree growth is controlled by a summability requirement formulated with the help of a generalized Randić index. In particular, it is proven that the Gibbs fields obey uniform integrability estimates, which are then used in the study of the topological properties of the set of Gibbs fields. In the second part, a class of graphs is introduced in which the mentioned summability is obtained by assuming that the vertices of large degree are located at large distances from each other. This is a stronger version of the metric property employed in Bassalygo and Dobrushin (1986).

Gibbs Random Fields with Unbounded Spins on Unbounded Degree Graphs

Gibbs fields are constructed and studied which correspond to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs of a certain type, for which the Gaussian Gibbs fields need not be existing. In these graphs, the vertex degree growth is controlled by a summability requirement formulated with the help of a generalized Randić index. In particular, it is proven that the Gibbs fields obey uniform integrability estimates, which are then used in the study of the topological properties of the set of Gibbs fields. In the second part, a class of graphs is introduced in which the mentioned summability is obtained by assuming that the vertices of large degree are located at large distances from each other. This is a stronger version of the metric property employed in Bassalygo and Dobrushin (1986).

Novel electrospun polyurethane/gelatin composite meshes for vascular grafts

Novel polymeric micro-nanostructure meshes as blood vessels substitute have been developed and investigated as a potential solution to the lack of functional synthetic small diameter vascular prosthesis. A commercial elastomeric polyurethane (Tecoflex EG-80A) and a natural biopolymer (gelatin) were successfully co-electrospun from different spinnerets on a rotating mandrel to obtain composite meshes benefiting from the mechanical characteristics of the polyurethane and the natural biopolymer cytocompatibility. Morphological analysis showed a uniform integration of micrometric (Tecoflex) and nanometric (gelatin) fibers. Exposure of the composite meshes to vapors of aqueous glutaraldehyde solution was carried out, to stabilize the gelatin fibers in an aqueous environment. Uniaxial tensile testing in wet conditions demonstrated that the analyzed Tecoflex-Gelatin specimens possessed higher extensibility and lower elastic modulus than conventional synthetic grafts, providing a closer matching to native vessels. Biological evaluation highlighted that, as compared with meshes spun from Tecoflex alone, the electrospun composite constructs enhanced endothelial cells adhesion and proliferation, both in terms of cell number and morphology. Results suggest that composite Tecoflex-Gelatin meshes could be promising alternatives to conventional vascular grafts, deserving of further studies on both their mechanical behaviour and smooth muscle cell compatibility.

Characterization and Structural Studies of the Plasmodium falciparum Ubiquitin and Nedd8 Hydrolase UCHL3

Like their human hosts, Plasmodium falciparum parasites rely on the ubiquitin-proteasome system for survival. We previously identified PfUCHL3, a deubiquitinating enzyme, and here we characterize its activity and changes in active site architecture upon binding to ubiquitin. We find strong evidence that PfUCHL3 is essential to parasite survival. The crystal structures of both PfUCHL3 alone and in complex with the ubiquitin-based suicide substrate UbVME suggest a rather rigid active site crossover loop that likely plays a role in restricting the size of ubiquitin adduct substrates. Molecular dynamics simulations of the structures and a model of the PfUCHL3-PfNedd8 complex allowed the identification of shared key interactions of ubiquitin and PfNedd8 with PfUCHL3, explaining the dual specificity of this enzyme. Distinct differences observed in ubiquitin binding between PfUCHL3 and its human counterpart make it likely that the parasitic DUB can be selectively targeted while leaving the human enzyme unaffected.

Rotating flows with acceleration and compaction in the model of hard spheres

We construct a model of the interaction between two flows in a gas of hard spheres, each of which has the structure of an accelerating and compacting rotating flow. We obtain several conditions sufficient for the uniform integral or the “mixed” norm of the difference between the left- and right-hand sides of the Boltzmann equation to be arbitrarily small.