Reversible Markov Chains

2019 ◽  
pp. 405-427
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Gaussian Approximation ◽  
Building Blocks ◽  
Partial Sums ◽  
Reversible Markov Chain ◽  
Maximal Inequalities ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.

On the invariance principle for reversible Markov chains

2016 ◽  
Vol 53 (2) ◽  
pp. 593-599 ◽  
Author(s):  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Stochastic Processes ◽  
Partial Sums ◽  
Additive Functionals ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

Abstract In this paper we investigate the functional central limit theorem (CLT) for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional CLT is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.

scholarly journals On the Functional Central Limit Theorem for Reversible Markov Chains with Nonlinear Growth of the Variance

2012 ◽  
Vol 49 (4) ◽  
pp. 1091-1105 ◽  
Author(s):  
Martial Longla ◽  
Costel Peligrad ◽  
Magda Peligrad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Central Limit ◽  
Adjoint Operator ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Conditional Convergence ◽  
Functional Clt ◽  
Functional Central Limit

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

On the Functional Central Limit Theorem for Reversible Markov Chains with Nonlinear Growth of the Variance

2012 ◽  
Vol 49 (04) ◽  
pp. 1091-1105
Author(s):  
Martial Longla ◽  
Costel Peligrad ◽  
Magda Peligrad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Central Limit ◽  
Adjoint Operator ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Conditional Convergence ◽  
Functional Clt ◽  
Functional Central Limit

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

Invariant measures of time-reversible Markov chains

1979 ◽  
Vol 16 (01) ◽  
pp. 226-229 ◽  
Author(s):  
P. Suomela
Keyword(s):  
Markov Chain ◽  
Invariant Measure ◽  
Markov Chains ◽  
Explicit Formula ◽  
Invariant Measures ◽  
Transition Probabilities ◽  
Time Reversibility ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.

Invariant measures of time-reversible Markov chains

1979 ◽  
Vol 16 (1) ◽  
pp. 226-229 ◽  
Author(s):  
P. Suomela
Keyword(s):  
Markov Chain ◽  
Invariant Measure ◽  
Markov Chains ◽  
Explicit Formula ◽  
Invariant Measures ◽  
Transition Probabilities ◽  
Time Reversibility ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.

Monotonicity of Positive Dependence with Time for Stationary Reversible Markov Chains

1995 ◽  
Vol 9 (2) ◽  
pp. 227-237 ◽  
Author(s):  
Taizhong Hu ◽  
Harry Joe
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Real Line ◽  
Continuous Time ◽  
Marginal Distribution ◽  
Random Vectors ◽  
Positive Dependence ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

Let (X1, X2) and (Y1, Y2) be bivariate random vectors with a common marginal distribution (X1, X2) is said to be more positively dependent than (Y1, Y2) if E[h(X1)h(X2)] ≥ E[h(Y1)h(Y2)] for all functions h for which the expectations exist. The purpose of this paper is to study the monotonicity of positive dependence with time for a stationary reversible Markov chain [X1]; that is, (Xs, Xl+s) is less positively dependent as t increases. Both discrete and continuous time and both a denumerable set and a subset of the real line for the state space are considered. Some examples are given to show that the assertions established for reversible Markov chains are not true for nonreversible chains.

Geometric Approaches to the Estimation of the Spectral Gap of Reversible Markov Chains

1993 ◽  
Vol 2 (3) ◽  
pp. 301-323 ◽  
Author(s):  
Salvatore Ingrassia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Spectral Gap ◽  
Systematic Treatment ◽  
Reversible Markov Chain ◽  
Underlying Graph ◽  
Reversible Markov Chains

In this paper we consider the problem of estimating the spectral gap of a reversible Markov chain in terms of geometric quantities associated with the underlying graph. This quantity provides a bound on the rate of convergence of a Markov chain towards its stationary distribution. We give a critical and systematic treatment of this subject, summarizing and comparing the results of the two main approaches in the literature, algebraic and functional. The usefulness and drawbacks of these bounds are also discussed here.

Functional Central Limit Theorem for Empirical Processes

2019 ◽  
pp. 428-437
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Empirical Process ◽  
Gaussian Approximation ◽  
Type Inequality ◽  
Stationary Sequence ◽  
General Criterion ◽  
Functional Clt ◽  
Finite Dimensional ◽  
Rosenthal Type Inequality ◽  
Mixing Sequences ◽  
Functional Central Limit

Here we discuss the Gaussian approximation for the empirical process under different kinds of dependence assumptions for the underlying stationary sequence. First, we state a general criterion to prove tightness of the empirical process associated with a stationary sequence of uniformly distributed random variables. This tightness criterion can be verified for many different dependence structures. For ρ‎-mixing sequences, by an application of a Rosenthal-type inequality, tightness is verified under the same condition leading to the usual CLT. For α‎-dependent sequences whose α‎-dependent coefficients decay polynomially to zero, it is shown to hold with the help of the Rosenthal inequality stated in Section 3.3. Since the asymptotic behavior of the finite-dimensional distributions of the empirical process is handled via the CLT developed in previous chapters, we then derive the functional CLT for the empirical process associated with the above-mentioned classes of stationary sequences. β‎-dependent sequences are also investigated by directly proving tightness of the empirical process.

scholarly journals Event-Chain Monte Carlo: Foundations, Applications, and Prospects

2021 ◽  
Vol 9 ◽  
Author(s):  
Werner Krauth
Keyword(s):  
Aqueous Solution ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Real World ◽  
Continuous Time ◽  
Statistical Physics ◽  
Reversible Markov Chains ◽  
Sampling Problem ◽  
Review Reports

This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply ECMC to the sampling problem in molecular simulation, i.e., to real-world models of peptides, proteins, and polymers in aqueous solution.

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