Perturbed Markov Chains with Damping Component

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

     

Relaxation of Product Markov Chains on Product Spaces

We study the relaxation time of product-type Markov chains approaching a product distribution. We bound the approach to stationarity for such Markov chains in terms of the mixing times of the component Markov chains. In cases where the component mixing times differ considerably we propose an optimized visiting scheme which makes such product-type Markov chains comparable to Gibbs-type samplers. We conclude the paper by discussing the relaxation of Metropolis-type samplers for separable energy functions.     

Imprecise Markov chains with absorption

We consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. We prove that under conditioning on non-absorption the imprecise conditional probabilities converge independently of the initial imprecise probability distribution if some regularity conditions are assumed. This is a generalisation of a known result from the classical theory of Markov chains by Darroch and Seneta [6].     

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.     

Remarks on the speed of convergence of mixing coefficients and applications

We study dependence coefficients for copula-based Markov chains. We provide new tools to check the convergence rates of mixing coefficients of copula-based Markov chains. We apply results to the Metropolis–Hastings algorithm. A necessary condition for symmetric copulas is given and mixtures of copulas are studied.     

一类特殊非齐次树上马氏链的若干强大数定律

首先给出了一类特殊非齐次树上可数状态马氏链的局部收敛定理,作为推论,得到了此类树上可数状态马氏链关于状态与状态序偶出现频率的若干极限性质,最后得到了这类特殊非齐次树上有限状态马氏链关于状态与状态序偶出现频率的强大数定律.     

Pattern discrete and mixed Hit-and-Run for global optimization

We develop new Markov chain Monte Carlo samplers for neighborhood generation in global optimization algorithms based on Hit-and-Run. The success of Hit-and-Run as a sampler on continuous domains motivated Discrete Hit-and-Run with random biwalk for discrete domains. However, the potential for efficiencies in the implementation, which requires a randomization at each move to create the biwalk, lead us to a different approach that uses fixed patterns in generating the biwalks. We define Sphere and Box Biwalks that are pattern-based and easily implemented for discrete and mixed continuous/discrete domains. The pattern-based Hit-and-Run Markov chains preserve the convergence properties of Hit-and-Run to a target distribution. They also converge to continuous Hit-and-Run as the mesh of the discretized variables becomes finer, approaching a continuum. Moreover, we provide bounds on the finite time performance for the discrete cases of Sphere and Box Biwalks. We embed our samplers in an Improving Hit-and-Run global optimization algorithm and test their performance on a number of global optimization test problems.     

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In this paper, we mainly studied the limit properties for the countable nonhomogeneous Markov chains. We established some limit properties for the functions of the countable nonhomogeneous Markov chains with variables under the convergence in the sense, which extended the similar conclusions for the functions with two variables. At last, as a corollary, we given the similar result in the homogeneous Markov stock market.     

渐近循环马氏链的收敛速度

引入了渐近循环马氏链的概念,它是循环马氏链概念的推广.首先研究了在强遍历的条件下,可列循环马氏链的收敛速度,作为主要结论给出了当满足不同条件时可列渐近循环马氏链的C-强遍历性,一致C-强遍历性和一致C-强遍历的收敛速度     

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.     

二重非齐次马氏链的遍历性及其应用

给出二重非齐次马氏链的强遍历性,绝对平均强遍历性,Cesaro平均收敛的概念．利用二维马氏链的遍历性和C-K方程,建立了二维马氏链与二重非齐次马氏链遍历性的关系．并讨论了齐次二重马氏链绝对平均强遍历与强遍历的等价性．最后给出Cesaro平均收敛在马氏决策过程和信息论中应用．     

双根树上二阶非齐次马氏链的强大数定律和Shannon-McMillan定理

本文首先研究了双根树上转移矩阵为逐点转移的二阶非齐次马氏链的强极限定理,同时得到双根树上二阶非齐次马氏链的强大数定律.最后,给出了双根树上二阶非齐次马氏链几乎处处收敛意义下的Shannon-McMillan定理.     

A Local Limit Theorem for a Family of Non-Reversible Markov Chains

By proving a local limit theorem for higher-order transitions, we determine the time required for necklace chains to be close to stationarity. Because necklace chains, built by arranging identical smaller Markov chains around a directed cycle, are not reversible, have little symmetry, do not have uniform stationary distributions, and can be nearly periodic, prior general bounds on rates of convergence of Markov chains either do not apply or give poor bounds. Necklace chains can serve as test cases for future techniques for bounding rates of convergence.     

m重非齐次马氏链的Cesaro平均收敛性

引入m重非齐次马氏链的Cesaro平均收敛的概念,给出并证明m重非齐次马氏链的一个Cesaro平均收敛定理.作为应用,得到了m重非齐次马氏链熵率存在的一个定理.     

On Markov Chain Approximations to Semilinear Partial Differential Equations Driven by Poisson Measure Noise

We consider the stochastic model of water pollution, which mathematically can be written with a stochastic partial differential equation driven by Poisson measure noise. We use a stochastic particle Markov chain method to produce an implementable approximate solution. Our main result is the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.     

一种无约束全局优化的水平值下降算法

本文研究无约束全局优化问题,建立了一种新的水平值下降算法(Level-value Descent Method,LDM).讨论并建立了概率意义下取全局最小值的一个充分必要条件,证明了算法LDM是依概率测度收敛的.这种LDM算法是基于重点度取样(Improtance Sampling)和Markov链Monte-Carlo随机模拟实现的,并利用相对熵方法(TheCross-Entropy Method)自动更新取样密度,算例表明LDM算法具有较高的数值精度和较好的全局收敛性.     

STRONG LAW OF LARGE NUMBERS AND SHANNON-MCMILLAN THEOREM FOR MARKOV CHAINS FIELD ON CAYLEY TREE

1 lntroductionA tr(t(t is a grapl1 G = {T, E) wllicl1 is co1l11ected a11d colltai1ls no cir(.uits. Give11 a11y twov(irtic'is 't / P E T, let crP bc tl1e ullique patl1 col111ecti11g,v aIld /]. Defille tl1e graph distal1cc(l(rr, p) to hc the Ilu1llber of edges co11tained in the path crP.We discuss ulainly tl1e rooted Cayley tree TC,2(i.e.,binary tree. See Fig.1). In tlle Cayleytree Tc,2,tl1e root (denoted by 0) llas OIlly two 11(tiglll)(irs al1d all otller vertices have threeneighbors. A…     

Convergence of difference additive functionals under local conditions on their characteristics

For additive functionals defined on a sequence of Markov chains that approximate a Markov process, we establish the convergence of functionals under the condition of local convergence of their characteristics (mathematical expectations).     

A new class of interacting Markov chain Monte Carlo methods

We present a new class of interacting Markov chain Monte Carlo methods to approximate numerically discrete-time nonlinear measure-valued equations. These stochastic processes belong to the class of self-interacting Markov chains with respect to their occupation measures. We provide several convergence results for these new methods including exponential estimates and a uniform convergence theorem with respect to the time parameter, yielding what seems to be the first results of this kind for this type of self-interacting models. We illustrate these models in the context of Feynman–Kac distribution semigroups arising in physics, biology and in statistics.     

Comparison of Three Web Search Algorithms

In this paper we discuss three important kinds of Markov chains used in Web search algorithms-the maximal irreducible Markov chain, the miuimal irreducible Markov chain and the middle irreducible Markov chain, We discuss the stationary distributions, the convergence rates and the Maclaurin series of the stationary distributions of the three kinds of Markov chains. Among other things, our results show that the maximal and minimal Markov chains have the same stationary distribution and that the stationary distribution of the middle Markov chain reflects the real Web structure more objectively. Our results also prove that the maximal and middle Markov chains have the same convergence rate and that the maximal Markov chain converges faster than the minimal Markov chain when the damping factor α 〉1/√2.