This useful property comes from the fact that, under a suitable transformation, the mpcn algorithm becomes a randomwalk metropolis algorithm. Meyny june 27, 2009 abstract we argue that the spectral theory of nonreversible markov chains may often be more e ectively cast within the framework of the naturally associated weightedl 1 space lv, instead of the usual hilbert space l 2 l. Exponential ergodicity for markov processes with random switching. Ergodicity of markov chain monte carlo with reversible. Ergodic properties of markov processes july 29, 2018 martin hairer lecture given at the university of warwick in spring 2006 1 introduction markov processes describe the timeevolution of random systems that do not have any memory. The extremal index can be expressed in terms of this random walk. A markov chain is geometrically ergodic if it converges to its invariant distribution at a geometric rate in total variation norm. The ergodic theory of markov processes in recent years has received quite substantial attention. We give a sufficient condition which simultaneously guarantees both versions are geometrically ergodic. Extremal indices, geometric ergodicity of markov chains.
Metropolishastings algorithm, markov chains operator, spec. Based on the general state space markov chain theory, we derive results about tran sience, ergodicity, geometric. In particular part i contains an extensive overview. Variance bounding and geometric ergodicity of markov chain. Geometric ergodicity and hybrid markov chains roberts, gareth and rosenthal, jeffrey, electronic communications in probability, 1997. Theorem 6 that under suitable conditions, hybrid chains will \inherit the geometric ergodicity of their constituent chains. Geometric ergodicity and the spectral gap of nonreversible markov chains i. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Roberts g o and rosenthal j s 1997 geometric ergodicity and. Geometric ergodicity and hybrid markov chains core. Carlo simulation algorithms, the socalled hybrid chains. Markov chains are fundamental stochastic processes that have many diverse applications. This paper gives an overview of recurrence and ergodicity properties of a markov chain. Introduction this paper deals with ergodic properties of markov chains produced by the hamiltonian or hybrid monte carlo method hmc, a technique for approx. Various notions of geometric ergodicity for markov chains on general state spaces exist. Exponential ergodicity for markov processes with random switching march 27, 20 bertrand cloez1, martin hairer2 1 lama, universit. We then restrict ourselves to markov chains which take values in a nite state space. Geometric ergodicity of metropolis algorithms sciencedirect. The wandering mathematician in previous example is an ergodic markov chain.
The discrete time mmap kph k1lcfsgpr queue and its. We describe the ergodic properties of some metropolishastings algorithms for heavytailed target distributions. The results lead to efficient algorithms for computing various performance measures at the level of individual types of customers. Ergodicity of stochastic processes and the markov chain. Meyn2 brown university and university of illinois consider the partial sums st of a realvalued functional ft of a markov chain t with values in a general state space. This paper deals with ergodic properties of markov chains produced by the hamiltonian or hybrid monte carlo method hmc, a technique for approximating high dimensional integrals through stochastic simulation duane et al. Subgeometric ergodicity analysis of continuoustime markov. Geometric ergodicity of the randomwalkbased metropolis algorithm on r k has previously been studied by roberts and tweedie 1996. Ergodic properties of markov processes martin hairer. Tuomins geometric ergodicity of markov chains 189 for the geometric convergence of hpf towards its limit nf. Chapter 17 graphtheoretic analysis of finite markov chains.
Geometric ergodicity and hybrid markov chains probability. As a corollary, we obtain a tintegrated form of geiii for geometrically ergodic chains. We will encounter this situation in section 5 for markov chains arising from the random walk metropolis algorithm. On the geometric ergodicity of hamiltonian monte carlo deepai. Citeseerx geometric ergodicity and hybrid markov chains. We investigate randomtime statedependent fosterlyapunov analysis on subgeometric rate ergodicity of continuoustime markov chains ctmcs. We then apply these results to a collection of chains commonly used in markov chain monte carlo simulation algorithms, the socalled hybrid chains. A markov chain is transient if all of its states are transient.
First, practical criteria for mg1type markov chains are obtained by analyzing the generating function of the first return probability to level 0. On geometric ergodicity of charme models request pdf. Markov chains and stochastic stability request pdf. This suggests the possibility of establishing the geometric ergodicity of large and complicated markov chain algorithms, simply by verifying the geometric ergodicity of the simpler chains which give rise to them. Within the class of stochastic processes one could say that markov chains are characterised by the dynamical property that they never look back. Inria geometric ergodicity in hidden markov models. A sufficient condition for geometric ergodicity of an ergodic markov chain is the doeblin condition see, for example, which for a discrete finite or countable markov chain may be stated as follows. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. Using those algorithms, the impact of the lcfsgpr service discipline on the corresponding queueing system can be analyzed. On the geometric ergodicity of hamiltonian monte carlo arxiv. This paper deals with ergodic properties of markov chains produced by the. Variable transformation to obtain geometric ergodicity in. Uniform geometric recurrence of markov decision chains. We argue that the spectral theory of nonreversible markov chains may often be more effectively cast within the framework of the naturally associated.
Chapter 1 markov chains a sequence of random variables x0,x1. A markov chain is called an ergodic or irreducible markov chain if it is possible to eventually get from every state to every other state with positive probability. Geometric ergodicity through variable transformation in. Geometric ergodicity of gibbs samplers in bayesian. In order to study geometric ergodicity for the process we start with the chain. Ergodicity theorem the foundation of markov chain theory is the ergodicity theorem. Geometric ergodicity for some spacetime maxstable markov.
We argue that the spectral theory of nonreversible markov chains may often be more effectively cast within the framework of the naturally associated weightedl. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The geometric ergodicity property, apart from theoretically ensuring convergence of the underlying markov chain to the stationary distribution at a geomet. The kendalls theorem and its application to the geometric ergodicity of markov chains witold bednorz institute of mathematics, warsaw university 02097 warszawa, poland email. Geometric ergodicity refers to the rate of convergence to the invariant distribution.
This observation is, in part, based on the following results. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned cranknicolson mpcn algorithm has geometric ergodicity even for heavytailed target distributions. Markov chain monte carlo mcmc algorithms such as the metropolis. Geyer school of statistics university of minnesota april, 11, 20 coauthor. Hastings chain is geometrically ergodic if the rejection probability of the chain is bounded away from unity. A homogeneous markov chain on a countable state space can be classified as ergodic, geometrically ergodic, or strongly ergodic.
It is instructive to consider what failure of one of the ergodicity conditions for markov chains means, because that can shed light on ergodicity more generally. The kendalls theorem and its application to the geometric. Recently dutta and bhattacharya 20 introduced a novel markov chain monte carlo methodology that can simultaneously update all the components of high dimensional parameters using simple deterministic transformations of a onedimensional random. We are mainly concerned with making use of the available results on deterministic statedependent drift conditions for ctmcs and on randomtime statedependent drift conditions for discretetime markov chains and transferring them to ctmcs. Rosenthal 1997, geometric ergodicity and hybrid markov chains. Strengthening ergodicity to geometric ergodicity for markov chains. In this paper we study polynomial and geometric exponential ergodicity for mg1type markov chains and markov processes. We present various sucient conditions implying v uniform ergodicity of the rsm when the target density decreases either subexponentially or exponentially in the tails.
However, since s 2 is a compact and metrisable space and 0. Uniform and geometric ergodicity under mixing and composition alicia a. A markov chain can be characterized by the properties of its states. This article attempts to build on the results of roberts and rosenthal 1997, which consider geometric ergodicity properties of hybrid chains in terms of their constituent component. Geometric ergodicity through variable transformation in metropolis random walk markov chain monte carlo charles j. On geometric and algebraic transience for discretetime.
Geometric ergodicity in the context of markov chains, ergodic means aperiodic. In this setting, we use a di erent method, as in, to prove a central limit theorem for functions of ergodic markov chains, where we have to impose. Several types of ergodicity for mg1type markov chains. We then apply these results to a collection of chains commonly used in markov chain monte carlo simulation algorithms, the socalled hybrid. Processes which are ergodic but not geometrically ergodic often require a more subtle treatment than we provide here. We then apply these results to a collection of chains commonly used in markov chain monte. Research is motivated by elegant mathematics as well as a range of applications. On geometric ergodicity of additive and multiplicative. We study the question of geometric ergodicity in a class of markov chains on the state space of nonnegative integers for which, apart from a finite number of boundary rows and columns, the elements pjk of the onestep transition matrix are of the form c kj where c k is a probability distribution on the set of integers. Geometric ergodicity for classes of homogeneous markov chains. Keywords geometric ergodicity uniform ergodicity markov chain monte carlo gibbs sampler metropoliswithingibbs random scan convergence rate citation johnson, alicia a jones, galin l neath, ronald c.
Chan, university ofchicago abstract it is known that if an irreducible and aperiodic, markov chain satisfies a drift condition in terms of a nonnegative measurable function gx, it is geometrically ergodic. Ergodicity of markov chain monte carlo with reversible proposal volume 54 issue 2 k. Thus, the most classical results about geometric ergodicity of markov chains, to be found e. It is common practice in markov chain monte carlo to update the simulation one variable or subblock of variables at a time, rather than conduct a single fulldimensional update. Variance bounding and geometric ergodicity of markov chain monte carlo kernels for approximate bayesian computation. Under mild assumptions on the coefficients of both the true and the assumed models, we prove that. Geometric ergodicity in a class of denumerable markov chains. In this case the transition probability graph of the chain is not strongly connected.
Spectrum of the metropolishastings chain with an application to. Linear programming and constrained optimal control of queues. Ergodicity and strong ergodicity have been characterized using the. Geometric ergodicity along with a moment condition results in the existence of a markov chain central limit theorem for.
Roberts g o and rosenthal j s 1997 geometric ergodicity and hybrid markov from math 4740 at cornell university. A note on the geometric ergodicity of a markov chain volume 21 issue 3 k. We consider an hidden markov model with multidimensional observations, and with misspecification, i. These criteria are given by, for example, theorems 4. Uniform ergodicity of the iterated conditional smc and geometric ergodicity of particle gibbs samplers andrieu, christophe, lee, anthony, and vihola, matti, bernoulli, 2018. On geometric ergodicity of additive and multiplicative transformation based markov chain monte carlo in high dimensions. First, suppose a chain fails to be positiverecurrent.
Pdf strengthening ergodicity to geometric ergodicity for. The paper deals with non asymptotic computable bounds for the geometric convergence rate of homogeneous ergodic markov processes. This is especially true because most instances of the metropolishastingsgreen algorithm are reversible or can be made to be reversible geyer, 2011, sections 1. Markov chain monte carlo, markov chains, stochastic simulation, hamiltonian dynamics, hamiltonian monte carlo, hybrid monte carlo, geometric ergodicity. A note on the geometric ergodicity of a markov chain. Uniform and geometric ergodicity under mixing and composition. Geometric ergodicity and the spectral gap of nonreversible markov chains article pdf available in probability theory and related fields 1541. Jun 29, 2009 geometric ergodicity and the spectral gap of nonreversible markov chains article pdf available in probability theory and related fields 1541 june 2009 with 70 reads how we measure reads. We study geometric ergodicity of deterministic and random scan versions of the twovariable gibbs sampler. With our focus on subgeometric ergodicity, which, loosely speaking, is a kind of convergence that is faster than ordinary ergodicity but slower than geometric ergodicity, much study is needed especially for continuous markov processes. This property is applied to nonparametric estimation in ergodic diffusion processes. On the central limit theorem for geometrically ergodic markov chains. Thus geometric ergodicity is a very desirable property for a markov chain to have.
Some sufficient conditions are stated for simultaneous geometric ergodicity of markov chain classes. Pdf geometric ergodicity and the spectral gap of non. The moment condition ee 2 t geometric ergodicity for charme models, which are generalized versions of the hidden markov model. S geometric ergodicity and hybrid markov chains, electr.