Convergence rates of symmetric scan Gibbs sampler

We consider the symmetric scan Gibbs sampler, and give some explicit estimates of convergence rates on the Wasserstein distance for this Markov chain Monte Carlo under the Dobrushin uniqueness condition.     

Data-driven risk-averse stochastic optimization with Wasserstein metric

In this paper, we study a data-driven risk-averse stochastic optimization approach with Wasserstein Metric for the general distribution case. By using the Wasserstein Metric, we can successfully reformulate the risk-averse two-stage stochastic optimization problem with distributional ambiguity to a traditional two-stage robust optimization problem. In addition, we derive the worst-case distribution and perform convergence analysis to show that the risk aversion of the proposed formulation vanishes as the size of historical data grows to infinity.     

An elementary proof of the triangle inequality for the Wasserstein metric

We give an elementary proof for the triangle inequality of the -Wasserstein metric for probability measures on separable metric spaces. Unlike known approaches, our proof does not rely on the disintegration theorem in its full generality; therefore the additional assumption that the underlying space is Radon can be omitted. We also supply a proof, not depending on disintegration, that the Wasserstein metric is complete on Polish spaces.

     

Geometric ergodicity of the Gibbs sampler for the Poisson change-point model

Poisson change-point models have been widely used for modelling inhomogeneous time-series of count data. There are a number of methods available for estimating the parameters in these models using iterative techniques such as MCMC. Many of these techniques share the common problem that there does not seem to be a definitive way of knowing the number of iterations required to obtain sufficient convergence. In this paper, we show that the Gibbs sampler of the Poisson change-point model is geometrically ergodic. Establishing geometric ergodicity is crucial from a practical point of view as it implies the existence of a Markov chain central limit theorem, which can be used to obtain standard error estimates. We prove that the transition kernel is a trace-class operator, which implies geometric ergodicity of the sampler. We then provide a useful application of the sampler to a model for the quarterly driver fatality counts for the state of Victoria, Australia.     

Duality in inhomogeneous random graphs,and the cut metric

The classical random graph model G(n, c/n) satisfies a “duality principle”, in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 399–411, 2011     

Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of certain Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of 1/4 with respect to every p-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than 1/4. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.     

Analysis of convergence rates of some Gibbs samplers on continuous state spaces

We use a non-Markovian coupling and small modifications of techniques from the theory of finite Markov chains to analyze some Markov chains on continuous state spaces. The first is a generalization of a sampler introduced by Randall and Winkler, and the second a Gibbs sampler on narrow contingency tables.     

Random walks in random environments on metric groups

Random walks in random environments on countable metric groups with bounded jumps of the walking particle are considered. The transition probabilities of such a random walk from a pointx εG (whereG is the group in question) are described by a vectorp(x) ε ℝ^|W| (whereW ⊏G is fixed and |W|<∞). The set {p(x),x εG} is assumed to consist of independent identically distributed random vectors. A sufficient condition for this random walk to be transient is found. As an example, the groups ℤ ^d , free groups, and the free product of finitely many cyclic groups of second order are considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 129–135, January, 2000.     

Convergence rates in the law of logarithm of random elements

1. IntroductionLet {Xu, n 2 1} be a sequence of r.v.IS in the same probability space and put Sa =nZ Xi, n 2 1; L(x) = mad (1, logx).i=1Since the definition of complete convergence is illtroduced by Hsu and Robbins[6], therehave been many authors who devote themselves to the study of the complete convergence forsums of i.i.d. real-valued r.v.'s, and obtain a series of elegys results, see [3,7]. Meanwhile,the convergence rates in the law of logarithm of i.i.d. real-vained r.v.'s have also be…     

Mixing time of an unaligned Gibbs sampler on the square

The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square ${[0,1]}^2$ to approach a stationary distribution with density proportional to $exp(?A^2{(u?v)}^2)$ for $(u,v)\in {[0,1]}^2$ with some large parameter $A$.Diaconis conjectured the mixing time of this process to be $O\left(A^2\right)$ which we confirm in this paper. This improves on the currently known $O(exp\left(A^2\right))$ estimate.     

On the Convergence of Successive Substitution Sampling

Abstract

The problem of finding marginal distributions of multidimensional random quantities has many applications in probability and statistics. Many of the solutions currently in use are very computationally intensive. For example, in a Bayesian inference problem with a hierarchical prior distribution, one is often driven to multidimensional numerical integration to obtain marginal posterior distributions of the model parameters of interest. Recently, however, a group of Monte Carlo integration techniques that fall under the general banner of successive substitution sampling (SSS) have proven to be powerful tools for obtaining approximate answers in a very wide variety of Bayesian modeling situations. Answers may also be obtained at low cost, both in terms of computer power and user sophistication. Important special cases of SSS include the “Gibbs sampler” described by Gelfand and Smith and the “IP algorithm” described by Tanner and Wong. The major problem plaguing users of SSS is the difficulty in ascertaining when “convergence” of the algorithm has been obtained. This problem is compounded by the fact that what is produced by the sampler is not the functional form of the desired marginal posterior distribution, but a random sample from this distribution. This article gives a general proof of the convergence of SSS and the sufficient conditions for both strong and weak convergence, as well as a convergence rate. We explore the connection between higher-order eigenfunctions of the transition operator and accelerated convergence via good initial distributions. We also provide asymptotic results for the sampling component of the error in estimating the distributions of interest. Finally, we give two detailed examples from familiar exponential family settings to illustrate the theory.     

A random weak ergodic property of infinite products of operators in metric spaces

ABSTRACT

We study the random weak ergodic property of infinite products of mappings acting on complete metric spaces. Our results describe an aspect of the asymptotic behaviour of random infinite products of such mappings. More precisely, we show that in appropriate spaces of sequences of operators there exists a subset, which is a countable intersection of open and everywhere dense sets, such that each sequence belonging to this subset has the random weak ergodic property. Then we show that several known results in the literature can be deduced from our general result.     

On the probability that finite spaces with random distances are metric spaces

For a set of 3 or 4 points we compute the exact probability that, after assigning the distances between these points uniformly at random from the set 1,…,n , the space obtained is metric. The corresponding results for random real distances follow easily. We also prove estimates for the general case of a finite set of points with uniformly random real distances.     

Hypothesis Tests of Convergence in Markov Chain Monte Carlo

Abstract

Deciding when a Markov chain has reached its stationary distribution is a major problem in applications of Markov Chain Monte Carlo methods. Many methods have been proposed ranging from simple graphical methods to complicated numerical methods. Most such methods require a lot of user interaction with the chain which can be very tedious and time-consuming for a slowly mixing chain. This article describes a system to reduce the burden on the user in assessing convergence. The method uses simple nonparametric hypothesis testing techniques to examine the output of several independent chains and so determines whether there is any evidence against the hypothesis of convergence. We illustrate the proposed method on some examples from the literature.     

Convergence rates in the central limit theorem for stationary mixing sequences of random vectors

Uniform and nonuniform Berry-Esseen bounds are given for strongly mixing and uniformly mixing stationary sequences of random vectors. The proofs are based on the classical Bernstein procedure.     

Convergence in law of random sums with non-random centering

We extend results obtained in Kruglov,⁽⁷⁾ and Finkelstein and Tucker⁽³⁾ to obtain necessary and sufficient conditions for convergence in law of random sums of non-identically distributed independent random variables under non-random centering. Thei.i.d. case is also considered for random variables attracted to a stable law. Necessary and sufficient conditions for convergence in law of these random variables under non-random centering, and in some cases, under non-random norming, are also obtained. The distribution functions for the limit laws are determined as well, generalizing results of Robbins.⁽¹⁰⁾ Supported in part by The State University of New York and United States Information Agency Grant No. IA AEMP69193692.     

Absolute continuity of Wasserstein geodesics in the Heisenberg group

In this paper we answer to a question raised by Ambrosio and Rigot [L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261-301] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the end-points is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolute continuity of Wasserstein geodesic also holds for Alexandrov spaces with curvature bounded from below.     

Convergence rates in the SLLN for some classes of dependent random fields

Let be a random field i.e. a family of random variables indexed by Nr, r?2. We discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions. Results are obtained for negatively associated random fields, ρ^?-mixing random fields (having maximal coefficient of correlation strictly smaller then 1) and martingale random fields.     

The diameter of a random subgraph of the hypercube

In this paper we present an estimation for the diameter of random subgraph of a hypercube. In the article by A. V. Kostochka (Random Struct Algorithms 4 (1993) 215–229) the authors obtained lower and upper bound for the diameter. According to their work, the inequalities n + m_p ≤ D(G_n) ≤ n + m_p + 8 almost surely hold as n → ∞, where n is dimension of the hypercube and m_p depends only on sampling probabilities. It is not clear from their work, whether the values of the diameter are really distributed on these 9 values, or whether the inequality can be sharpened. In this paper we introduce several new ideas, using which we are able to obtain an exact result: D(G_n) = n + m_p (almost surely). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012