Topology Of Weak Convergence Research Articles

The maximum domain of attraction of multivariate extreme value distributions is small

Consider the set of Borel probability measures on Rk and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.

Existence of flows for linear Fokker\u2013Planck\u2013Kolmogorov equations and its connection to well-posedness

Let the coefficients a_{ij} and b_i, i,j le d, of the linear Fokker–Planck–Kolmogorov equation (FPK-eq.) ∂tμt=∂i∂j(aijμt)-∂i(biμt)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t\\mu _t = \\partial _i\\partial _j(a_{ij}\\mu _t)-\\partial _i(b_i\\mu _t) \\end{aligned}$$\\end{document}be Borel measurable, bounded and continuous in space. Assume that for every s in [0,T] and every Borel probability measure nu on mathbb {R}^d there is at least one solution mu = (mu _t)_{t in [s,T]} to the FPK-eq. such that mu _s = nu and t mapsto mu _t is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution mu ^{s,nu } for each pair (s,nu ) such that this family of solutions fulfills μts,ν=μtr,μrs,νfor all0≤s≤r≤t≤T,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mu ^{s,\\nu }_t = \\mu ^{r,\\mu ^{s,\\nu }_r}_t \\text { for all }0 \\le s \\le r \\le t \\le T, \\end{aligned}$$\\end{document}which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unique if and only if the FPK-eq. is well-posed.

Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process.

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^*$. The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.

Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities

We consider the stochastic ranking process with space-time dependent unbounded jump rates for the particles. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution in the infinite particle limit. We assume topology of weak convergence for the space of distributions, which implies that the fluctuations among particles with different jump rates cancel in the limit. The results are proved by first finding an auxiliary stochastic ranking process, for which a strong law of large numbers is applied, and then applying a multi time recursive Gronwall's inequality. The limit has a representation in terms of non-Markovian processes which we call point processes with last-arrival-time dependent intensities.

A characterisation of isometries with respect to the Lévy-Prokhorov metric

According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L{\'e}vy--Prokhorov distance. Motivated by these facts, the famous Banach--Stone theorem, and some recent works related to characterisations of onto isometries of spaces of Borel probability measures, here we give a complete description of surjective isometries with respect to the L{\'e}vy--Prokhorov metric in case when the underlying metric space is a separable Banach space. Our result can be considered as a generalisation of L. Moln\'ar's earlier Banach--Stone-type result which characterises onto isometries of the space of all probability distribution functions on the real line wit respect to the L{\'e}vy distance. However, the present more general setting requires the development of an essentially new technique.

Large deviations for empirical measures of mean-field Gibbs measures

In this paper, we show that the empirical measure of mean-field model satisfies the large deviation principle with respect to the weak convergence topology or the stronger Wasserstein metric, under the strong exponential integrability condition on the negative part of the interaction potentials. In contrast to the known results we prove this without any continuity or boundedness condition on the interaction potentials. The proof relies mainly on the law of large numbers and the exponential decoupling inequality of de la Peña for U-statistics.

Probability Measures on Product Spaces with Uniform Metrics

For a countable product of complete separable metric spaces, with a topology induced by a uniform metric, the σ-algebra generated by the open balls, which was introduced by Dudley (1966), coincides with the product σ-algebra. Any probability measure on the product space with this σ-algebra is quasi-separable in the sense that, for any union of open balls that has full measure, there is a countable sub-union that also has full measure. With suitably adapted definitions, the topology of weak convergence on the space of such measures is equivalent to the topology induced by the Prohorov metric. The projection mapping from such measures to sequences of measures on the first l factors, l = 1; 2; ...; is a homeomorphism if the range of this mapping is also given a uniform metric. These findings are relevant for the theory of games of incomplete information, where a topology on the space of belief hierarchies that is based on a uniform metric has been proposed as being more appropriate for capturing the continuity properties of strategic behaviour.

Convex Analysis in Decentralized Stochastic Control, Strategic Measures, and Optimal Solutions

This paper is concerned with the properties of the sets of strategic measures induced by admissible team policies in decentralized stochastic control and the convexity properties in dynamic team problems. To facilitate a convex analytical approach, strategic measures for team problems are introduced. Properties such as convexity, compactness, and Borel measurability under weak convergence topology are studied, and sufficient conditions for each of these properties are presented. These lead to existence of and structural results for optimal policies. It will be shown that the set of strategic measures for teams which are not classical is in general nonconvex, but the extreme points of a relaxed set consist of deterministic team policies, which lead to their optimality for a given team problem under an expected cost criterion. Externally provided independent common randomness for static teams or private randomness for dynamic teams do not improve the team performance. The problem of when a sequential team p...

Convergent sequences of sparse graphs: A large deviations approach

Models based on sparse graphs are of interest to many communities: they appear as basic models in combinatorics, probability theory, optimization, statistical physics, information theory, and more applied fields of social sciences and economics. Different notions of similarity (and hence convergence) of sparse graphs are of interest in different communities. In probability theory and combinatorics, the notion of Benjamini-Schramm convergence, also known as left-convergence, is used quite frequently. Statistical physicists are interested in the the existence of the thermodynamic limit of free energies, which leads naturally to the notion of right-convergence. Combinatorial optimization problems naturally lead to so-called partition convergence, which relates to the convergence of optimal values of a variety of constraint satisfaction problems. The relationship between these different notions of similarity and convergence is, however, poorly understood. In this paper we introduce a new notion of convergence of sparse graphs, which we call Large Deviations or LD-convergence, and which is based on the theory of large deviations. The notion is introduced by “decorating” the nodes of the graph with random uniform i.i.d. weights and constructing corresponding random measures on [0,1] and [0,1]2. A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on [0,1]d,d=1,2. The corresponding large deviations rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object. We then establish that LD-convergence implies the other three notions of convergence discussed above, and at the same time establish several previously unknown relationships between the other notions of convergence. In particular, we show that partition-convergence does not imply left- or right-convergence, and that right-convergence does not imply partition-convergence. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Directed Information on Abstract Spaces: Properties and Variational Equalities

Directed information or its variants are utilized extensively in the characterization of the capacity of channels with memory and feedback, nonanticipative lossy data compression, and their generalizations to networks. In this paper, we derive several functional and topological properties of directed information, defined on general abstract alphabets (complete separable metric spaces), using the topology of weak convergence of probability measures. These include the convexity of the set of consistent distributions, which uniquely define causally conditioned distributions, convexity, and concavity of directed information with respect to the sets of consistent distributions, weak compactness of such sets of distributions, their joint distributions, and their marginals. Furthermore, we show lower semicontinuity of directed information, and under certain conditions, we also establish continuity. Finally, we derive variational equalities for directed information, including sequential versions. These may be viewed as the analog of the variational equalities of mutual information (utilized in Blahut–Arimoto algorithms). In summary, we extend the basic functional and topological properties of mutual information to directed information. These properties are discussed throughout this paper, in the context of extremum problems of directed information.

Brownian occupation measures, compactness and large deviations

In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_{t}(A)=\frac{1}{t}\int_{0}^{t}{\mathbf{1}}_{A}(W_{s})\,\mathrm{d}s$ of the $d$-dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $\mathcal{M}_{1}(\mathbb{R}^{d})$ can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^{d}$ by adding a point at $\infty$ that results in the compactification of $\mathcal{M}_{1}(\mathbb{R}^{d})$ by allowing some mass to escape to the point at $\infty$. If one were to use only test functions that are continuous and vanish at $\infty$, then the compactification results in the space of sub-probability distributions $\mathcal{M}_{\le1}(\mathbb{R}^{d})$ by ignoring the mass at $\infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $\tilde{\mathcal{M}}_{1}=\tilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ under the action of the translation group $\mathbb{R}^{d}$ on $\mathcal{M}_{1}(\mathbb{R}^{d})$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.

A Note on Stability for Risk-Averse Stochastic Complementarity Problems

In this paper, we propose a new approach to stochastic complementarity problems which allows to take into account various notions of risk aversion. Our model enhances the expected residual minimization formulation of Chen and Fukushima (Math Oper Res 30:1022---1038, 2005) by replacing the expectation with a more general convex, nondecreasing and law-invariant risk functional. Relevant examples of such risk functionals include the Expected Excess and the Conditional Value-at-Risk. We examine qualitative stability of the resulting one-stage stochastic optimization problem with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the decision vector and the entering probability measure. By a classical result from parametric optimization, this implies upper semicontinuity of the optimal value function. Throughout the analysis, we assume for building the model that a nonlinear complementarity function fulfills a certain polynomial growth condition. We conclude the paper demonstrating that this assumption holds in the vast majority of all practically relevant cases.

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence ξ˜1,ξ˜2,…, are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution e˜n:=1n∑i=1nδξ˜i of the n past observations, while the latter is compared to the m-fold product e˜nm. Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts δe˜n and e˜nm, respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the ξ˜n's, and not under hypothetical product laws p0∞, as p0 varies among the admissible determinations of a random probability measure.

Stein's method and a quantitative Lindeberg CLT for the Fourier transforms of random vectors

We use a multivariate version of Stein's method to establish a quantitative Lindeberg CLT for the Fourier transforms of random N-vectors. We achieve this, conceptually mainly by constructing a natural approach structure on N-random vectors overlying the topology of weak convergence, and technically mainly by deducing a specific integral representation for the Hessian matrix of a solution to the Stein equation with test function et(x)=exp⁡(−i∑k=1Ntkxk), where t,x∈RN.

Some Functional Large Deviations Principles in Nonparametric Function Estimation

In this paper, we investigate functional large deviation behaviors of some nonparametric function estimates. As a first step, we define a a vector process W n and study its large deviation behavior in the space L 1×L 1×L 1 with respect to the weak convergence topology. As by-products, we derive large deviation principles in the L 1-space equipped with the weak convergence topology simultaneously for several density and regression estimators built up using the delta-sequence estimation method.

Sanov’s theorem in the Wasserstein distance: A necessary and sufficient condition

Let ( X n ) n ≥ 1 be a sequence of i.i.d.r.v.’s with values in a Polish space ( E , d ) of law μ . Consider the empirical measures L n = 1 n ∑ k = 1 n δ X k , n ≥ 1 . Our purpose is to generalize Sanov’s theorem about the large deviation principle of L n from the weak convergence topology to the stronger Wasserstein metric W p . We show that L n satisfies the large deviation principle in the Wasserstein metric W p ( p ∈ [ 1 , + ∞ ) ) if and only if ∫ E e λ d p ( x 0 , x ) d μ ( x ) < + ∞ for all λ > 0 , and for some x 0 ∈ E .

Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems. We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology, while it does not satisfy the large deviation principles in the τ-topology.

Narrow Convergence in Spaces of Set-Valued Measures

Summary. We prove an analogue of Topsoe's criterion for relative compactness of a family of probability measures which are regular with respect to a family sets. We consider measures whose values are compact convex sets in a locally convex linear topological space. Introduction. Let T be an abstract set, K a family of subsets of T, and (E;F) a dual pair of real vector spaces, with E endowed with the weak topology (E;F). Let cc(E;F) be the set of all convex compact non-empty subsets of E, and f M+(T;K;cc(E;F)) the set of K-inner regular positive set-valued measures defined on a -fieldB of subsets of T and with values in cc(E;F). We denote by M+(T;K) the set ofK-inner regular non-negative measures defined on B provided with the topology of weak convergence. Prokhorov (11) has proved that if T is a Polish space andB the set of Borel subsets, then the relatively compact subsets of M+(T;K) are precisely the tight ones. But this result is not valid for all topological space (see e.g. (5), (10), (18)). In (16) Topsoe has characterized the relatively compact subsets of M+(T;K) in general situations. Before and after Topsoe's paper there were others (e.g. (1), (3), (18), (5)-(10)). In this paper we generalize to the space f

Uniform [formula omitted] posterior consistency in compact Gaussian shift experiments

For a suitably chosen metric ρ for the topology of weak convergence in the space of prior distributions on the shift parameter of a compact Gaussian shift experiment, the posterior distribution (induced by a full support prior) of the shift parameter given independent and identically distributed observations from the experiment is uniformly (in the shift parameter) L 1 consistent in ρ . The corresponding posterior mean is uniformly L 1 consistent in the norm on the parameter space. ρ -distance between the posterior mean (induced by a full support hyperprior) of the prior (i.e., mixing) distribution given independent and identically distributed observations from the mixture experiment and the empirical distribution of the parameter sequence in the product experiment [ the mixing distribution] goes to zero in L 1 of the product [mixture] experiment, uniformly in parameter sequences [mixing distributions].

Asymptotical behavior of linear congruential generators

The paper is devoted to asymptotical analysis of Linear Congruential Generators (LCGs) with increasing moduli and multiplicators. The study is performed within the scope of a stochastic model of LCGs, the technique of weak convergence of distributions is used. It is proved, f.e., that the classical spectral test can be described in terms of a special probability metric inducing a topology of weak convergence. A number of theoretical examples of LCGs with good and bad asymptotical equidistribution properties is presented.