Meyn3 abstractit is known that state dependent, multistep lyapunov bounds lead to greatly simpli. Randomtime state dependent stochastic drift criteria are presented together with a class of application areas in networked control systems. We show that these concepts of stability are largely equivalent for a major class of chains chains with continuous components, or if the state space has a sufficiently rich class of appropriate sets petite sets. Convergence rate of markov chains will perkins april 16, 20. Using the convex analytic approach under mild conditions, we prove that the optimal values and optimal policies of the original dtmdps. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in connor and kendall 2007 2. Convergence of probability measures and markov decision. Markov chains null space convergence time minimal polynomial leading vectors eigenvalues accessibility markov decision problem 1. Markov chains and stochastic stability second edition meyn and tweedie is back. Lyapunov analysis for rates of convergence in markov chains and randomtime statedependent drift m. On rates of convergence for markov chains under random time. Introduction one of the most widely discussed properties of markov chains is its convergence to a steady state, independently of the initial distribution.
It is named after the russian mathematician andrey markov. The spectral gap and perturbation bounds for reversible continuoustime markov chains journal of applied probability 414. On rates of convergence for markov chains under random time state dependent drift criteria ramiro zurkowski, serdar yu. This means that there is a possibility of reaching j from i in some number of steps. Oct 12, 2017 difference between recurrent state and transient state in markov chain i a state i is called recurrent, if we go from that state to any other state j, then there is at least one path to return back to i. Statedependent fosterlyapunov criteria for subgeometric convergence of markov chains. Transition kernel of a reversible markov chain 18 3. Randomtime, state dependent stochastic drift for markov chains and application to stochastic stabilization over erasure channels1 serdar yu. Abstractwe consider a form of statedependent drift condition for a general markov chain, whereby the chain subsampled at some deterministic time satisfies a. This paper is concerned with the convergence of a sequence of discretetime markov decision processes dtmdps with constraints, state action dependent discount factors, and possibly unbounded costs. The results suggest a simple rule for identifying the singular matrices which do not have a finite convergence time. It is named after the russian mathematician andrey markov markov chains have many applications as statistical models of realworld processes.
An absorbing state is a state that is impossible to leave once reached. The classical theorem of perron and frobenius which can be found for example in seneta 43 can be used to show that for. Convergence of markov decision processes with constraints and. In continuoustime, it is known as a markov process. Although our work will mostly be confined to working with petite sets, we know that for irreducible and aperiodic processes petite sets are in fact small sets 5. State space s a,c g t transition probabilities taken to be the observed frequencies a c g t a 0.
But thats another matter entirely from the usual notion of steady state for markov chains. This paper is concerned with the convergence of a sequence of discretetime markov decision processes dtmdps with constraints, stateaction dependent discount factors, and possibly unbounded costs. Convergence of probability measures and markov decision models with incomplete information eugene a. We say that i communicates with j written i j if i j and j i. Perhaps your best bet is to take a look at one of the bibles on this topic. To be considered a proper markov chain, a system must have a set of distinct states, with identifiable transitions between them. Take a look at wikipedias article on markov chains and specifically the notion of a steadystate distribution or stationary distribution, or read about the subject in your favorite textbook. Lyapunov analysis for rates of convergence in markov chains. This paper is concerned with theconvergence of a sequence of discretetime markov decisionlinebreak processes dtmdps with constraints, stateaction dependent discount factors, and possibly unbounded linebreak costs. Convergence last class we saw that if x n is an irreducible, aperiodic, positive recurrent markov chain, then there exists a stationary distribution on the state space x, so that no matter where the chain starts, x n. Stochastic stability and drift criteria for markov chains.
On january 23, 19, he summarized his findings in an address to the imperial. We investigate subgeometric rate ergodicity for markov chains in the wasserstein metricand show that the finiteness of the expectation ei,j. On state dependent criteria for stability, traditionally it is assumed that the stopping times take the form where is a deterministic function of the state. Thanks for contributing an answer to mathematics stack exchange. Various results exist for subgeometric and geometric ergodicity with different techniques. Zgurovskyb received june 2014 abstractthis paper deals with three major types of convergence of probability measures on metric spaces. However, as far as i can tell, there are minor differences in the results presented in the paper and the book, and the paper do have a proof of theorem 4. A markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. In this paper we survey approaches to studying the ergodicity of aperiodic and irreducible markov chains 3, 18, 5, 12, 19. Statedependent criteria for convergence of markov chains article pdf available in the annals of applied probability 41 february 1994 with 34 reads how we measure reads. Cross validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Markov chains simple examples simple examples of dna sequence modeling a markov chain model for the dna sequence shown earlier. Convergence of markov model computer science stack exchange. Difference between recurrent state and transient state in markov chain i a state i is called recurrent, if we go from that state to any other state j, then there is at least one path to return back to i.
This chapter presents an overview of the theory of markov chains and drift criteria to establish stochastic stability of markov chains. Convergence to equilibrium means that, as the time progresses, the markov chain forgets about its initial. Markov chains have many applications as statistical models. We are mainly concerned with making use of the available results on deterministic state dependent drift conditions for ctmcs and on randomtime state dependent drift conditions for discretetime markov chains and transferring them to ctmcs. This means that there is a possibility of reaching j from i in some. Speed of convergence to stationarity for stochastically. Basic markov chain theory to repeat what we said in the chapter 1, a markov chain is a discretetime. Probability of a timedependent set of states in markov chain. Fortb a department of mathematics, university of york, york, yo10 5dd, uk b ltci, cnrstelecom paristech, 46 rue barrault, 75634 paris c. Recent work by athreya and ney and by nummelin on the limit theory for markov chains shows that the close connection with regeneration theory holds also for chains on a general state space. The relation partitions the state space into communicating classes. As we discuss now, these two tasks are related to determining convergence of the underlying markov chain to stationarity and convergence of monte carlo estimators to population quantities, respectively. Statedependent fosterlyapunov criteria for subgeometric convergence of markov chains s. A note on subgeometric rate convergence for ergodic markov.
We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in connor and kendall 2007 concerning the existence of socalled tame. We investigate randomtime state dependent fosterlyapunov analysis on subgeometric rate ergodicity of continuoustime markov chains ctmcs. Another method would be to augment the state space by adding a state 0 that is a trapping state from which, if entered, you can never leave. This process takes place on some agreedon set called the state space, and often denoted by.
Although the general convergence theorem states that any finite markov chain that is aperiodic and irreducible converges to its stationary distribution. Rate of convergence of the ehrenfest random walk 23 1. Convergence diagnostic of markov chain that converge to uniform. Abstractwe consider a form of statedependent drift condition for a general markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric fosterlyapunov condition. For the rest of this thesis we only deal with time homogeneous markov chains, and use the notation fx tgfor markov chains and pas the one step transition probability. Minorization conditions and convergence rates for markov. Markov chains and random walks on graphs applying the same argument to at, which has the same. We consider a form of statedependent drift condition for a general markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric fosterlyapunov condition. A markov chain with transition kernel px,dy on a state space x is said to satisfy a minorization condition or split on a subset r. Statedependent fosterlyapunov criteria for subgeometric. It is known that under some standard conditions on the markov chain, for any initial value, the distribution of x. When a markov chain converges to a steady state, what kind of convergence is it. Markov chains and randomtime statedependent drift by ramiro a. We study the properties of finite ergodic markov chains whose transition probability matrix p is singular.
We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in connor and kendall 2007. Request pdf state dependent fosterlyapunov criteria for subgeometric convergence of markov chains we consider a form of state dependent drift condition for a general markov chain, whereby. The standard fosterlyapunov approach to establishing recurrence and ergodicity of markov chains requires that the onestep mean drift of the chain be negative outside some appropriately finite set. Naturally one refers to a sequence 1k 1k 2k 3 k l or its graph as a path, and each path represents a realization of the markov chain. Connor, gersende fort submitted on 16 jan 2009 v1, last revised 3 sep 2009 this version, v2. The probabilities of future events depend on the current state of the system. State dependent fosterlyapunov criteria for subgeometric convergence of markov chains s.
Journal of probability and statistics 2014 article. First links in the markov chain american scientist. Statedependent fosterlyapunov criteria for subgeometric convergence of markov chains article in stochastic processes and their applications 11912. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in connor and kendall 2007 concerning the existence of socalled tame markov chains. We investigate randomtime statedependent fosterlyapunov analysis on subgeometric rate ergodicity of continuoustime markov chains ctmcs. Markov founded a new branch of probability theory by applying mathematics to poetry.
Expected value and markov chains aquahouse tutoring. Pdf statedependent criteria for convergence of markov. Lyapunov analysis for rates of convergence in markov. Then appropriately adjust the transition probabilities. Delving into the text of alexander pushkins novel in verse eugene onegin, markov spent hours sifting through patterns of vowels and consonants. The authoritative reference on these matters is meyn and tweedies book markov chains and stochastic stability, which is also cited heavily in the paper. Here this is used to study extremal behaviour of stationary or asymptotically stationary markov chains. Markov chains and markov models university of helsinki. We quantify how the rate of ergodicity, nature of lyapunov functions, their drift properties, and the distributions of stopping times are related. Irreducibility is simple to check its equivalent to connectedness in graphs, and periodicity is also easy to check the definition of both is found in the first chapter of the book below. Markov chains with finite convergence time sciencedirect. Randomtime, statedependent stochastic drift for markov. Convergence of markov decision processes with constraints. Markov chains convergence theorems jeffrey rosenthal.
Lyapunovtheoretic drift criteria, we establish both subgeometric and geometric rates of convergence for markov chains under state dependent random time drift criteria. Orientation finitestate markov chains have stationary distributions, and irreducible, aperiodic. Class structure we say that a state i leads to j written i j if it is possible to get from i to j in some. What is the difference between a recurrent state and a. X, if there is a probability measure q on x, a positive integer k 0, and 0, such that. Mostly this project is based on the article general state space markov chains and mcmc algorithms by gareth o. The bible on markov chains in general state spaces has been brought up to date to re.
Basic markov chain theory to repeat what we said in the chapter 1, a markov chain is a discretetime stochastic process x1, x2. We call the state space irreducible if it consists of a. Stochastic stability and drift criteria for markov chains in. State dependent fosterlyapunov criteria for subgeometric convergence of markov chains authors. Probability of a timedependent set of states in markov. Roberts and rosenthal 19 show using the coupling inequality and nummelins splitting technique how geometric ergodicity follows from a simple drift condition. On rates of convergence for markov chains under random.
The results establish bounds on the convergence time of p m to a matrix where all the rows are equal to the stationary distribution of p. The spectral gap and perturbation bounds for reversible continuoustime markov chains journal of. Using the convex analytic approach under mild conditions, we prove that the optimal values and optimal policies of the original dtmdps converge to those of the limit one. Stability and exponential convergence of continuoustime markov chains journal of applied probability 404. Using the convex analytic approach under mild conditions, we prove that the optimal values and optimal policies of the original dtmdps converge to those of. Subgeometric ergodicity under randomtime statedependent. Malyshev and mensikov developed a refinement of this approach for countable state space chains, allowing the drift to be negative after a number. Well discuss conditions on the convergence of markov chains, and consider the proofs of convergence theorems in details. 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. Pdf statedependent criteria for convergence of markov chains. Longrun proportions convergence to equilibrium for irreducible, positive recurrent, aperiodic chains. Convergence last class we saw that if x n is an irreducible, aperiodic, positive recurrent markov chain, then there exists a stationary distribution on the state space x, so that no matter where the chain starts.
Convergence diagnostics for markov chain monte carlo. We consider a form of state dependent drift condition for a general markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric fosterlyapunov condition. These include tightness on the one hand and harris recurrence and ergodicity on the other. Maxima and exceedances of stationary markov chains. Bounds on the rate of convergence for one class of.
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