Citeseerx document details isaac councill, lee giles, pradeep teregowda. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of. Their wide readership includes leading researchers in the many fields. A nonstationary markov chain is weakly ergodic if the dependence of the state. Ratio ergodicity for nonhomogeneous markov chains in general.
This allows us to investigate the weak ergodicity of nonhomogeneous discrete markov processes ndmp by means of the ergodicity coefficient. The non homogeneous case is generally called time inhomogeneous or non stationary in time. Ergodic properties of nonhomogeneous markov chains defined on. Ratio ergodicity for nonhomogeneous markov chains in. Nonhomogeneous markov chains with desired properties the. A geometric approach to ergodic nonhomogeneous markov chains. Hence, we can use any tau coefficient to establish the weak ergodicity of a nonhomogeneous markov chain. Generalized dobrushin ergodicity coefficient and ergodicities of.
Further in section 4, we find necessary and sufficient conditions for the weak ergodicity of nonhomogeneous discrete markov chains ndmc, which extend the results of 22, 24 to an abstract. It is named after the russian mathematician andrey markov markov chains have many applications as statistical models of realworld processes. Ams proceedings of the american mathematical society. Seneta, explicit forms for ergodicity coefficients and spectrum localization. In this paper we intend to show that using the ideas and results of birkhoff and baueret al. We provide a sufficient conditions for such processes to satisfy the weak ergodicity. Nonhomogeneous markov chains, doeblins coe cient 1 introduction there is a large literature on the asymptotic behaviour of nonhomogeneous markov chains. This may occur in non homogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases. A vector norm b on c is absolute if bx bxl for anyx e.
The main object of our interest is a timehomogeneous markov process x with the. Weak ergodicity in nonhomogeneous markov chains mathematical. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Convergence of some time inhomogeneous markov chains via spectral techniques. Ergodic properties of nonhomogeneous markov chains defined. For then the determinant of each matrix is less than one in absolute value. Consensus of complex network systems and ergodicity of. The markov property is common in probability models because, by assumption, one supposes that the important variables for the system being modeled are all included in the state space. On products of nonnegative matrices cohn, harry and nerman, olle, the annals of probability, 1990.
Seneta, on the historical development of the theory of finite inhomogeneous markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered banach space with. A simple consequence of the theorem for stochastic matrices is the following. Minimal information with scalar normalization 80 c. Markov chains the concepts of ergodicity, strong ergodicity, and weak ergodicity coincide. A very mild sufficient condition of the consensus reaching that allows the communication among agents to be timedependent and directed is obtained by estimating the. Central limit theorem for nonstationary markov chains. Applications treat nearcritical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. We note that the ergodicity of this chain, and of analogous non homogeneous chains, can not be demonstrated by use of the sufficient conditions given by hamal 2 and mott 4. This paper deals with the ergodicity convergence of the marginals and. Let 1 1 1 1 22 22 1 1 1 1 q 22 andr 22 1 1 1 1 2 2 22.
State classification of timenonhomogeneous markov chains. Shen, a geometric approach to ergodic nonhomogeneous markov chains, in wavelet analysis and multiresolution methods, ed. Local stationarity and timeinhomogeneous markov chains. We now formally describe hidden markov models, setting the notations that will be used throughout the book. But, as will be shown later, for infinite chains, the notions of ergodicity and strong ergodicity are separated. Weak ergodicity of nonhomogeneous markov chains on noncommutative l1spaces. On the historical development of the theory of finite. Two approaches to the construction of perturbation bounds for.
It is known that the dobrushins ergodicity coefficient is one of the effective tools to study the behavior of nonhomogeneous markov chains. Subsequently, attention focussed rather on a particular type of nonhomogeneous markov chain, called weakly ergodic. Generalized dobrushin ergodicity coefficient and ergodicities of nonhomogeneous markov chains authors. Consensus of complex network systems and ergodicity of markov. Under mcmc, the markov chain is used to sample from some target distribution. 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 continuoustime, it is known as a markov process. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous markov chains. Weak ergodicity 1 the most crucial property of a markov chain is the con. It would be desirable to have the hypothesis of the theorem require. Unless stated to the contrary, all markov chains considered in these notes are time homogeneous and therefore the subscript l is omitted and we simply represent the matrix of transition probabilities as p p ij. Two approaches to the construction of perturbation bounds.
A vector norm b on c is absolute if bx bxl for anyx e c, where ix denotes the vector xi. All textbooks and lecture notes i could find initially introduce markov chains this way but then quickly restrict themselves to the time homogeneous case where you have one transition matrix. Conditions for weak ergodicity of inhomogeneous markov chains. Local stationarity and timeinhomogeneous markov chains 5 definition 1.
Discretetime discretestate random markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. Inspired by the work of daubechies and lagarias on a set of matrices with convergent infinite products, we study the geometric approach to the classical problem of weakly ergodic nonhomogeneous markov chains. Weak ergodicity of nonhomogeneous markov chains distributed control jadbabaei2003 flocking problem distributed computing 1983. Stochastic matrixvalued cocycles and nonhomogeneous markov.
Conversely, if x is a nonnegative random variable with a continuous distribution such that the conditional distribution of x. However, if the products of matrices are formed in the backward direction, i. Im trying to find out what is known about timeinhomogeneous ergodic markov chains where the transition matrix can vary over time. Asymptotic behaviour of random tridiagonal markov chains. We note that the ergodicity of this chain, and of analogous nonhomogeneous chains, can not be demonstrated by use of the sufficient conditions given by hamal 2 and mott 4. Stochastic matrixvalued cocycles and nonhomogeneous. Corollary 1 suppose a is a primitive stochastic matrix. Generalized dobrushin ergodicity coefficient and ergodicities of non homogeneous markov chains authors. In terms of this coefficient we prove the equivalence uniform.
It is shown that this consensus reaching is equivalent to the corresponding weak ergodicity of the markov process. Paz 1970, madsen 1971 and iosifescu 1972, who make use of the ergodic coefficient introduced by dobrusin 1956 in section 2. He, lecture notes in pure and applied mathematics, 212, pp. The following example illustrates some further points. A poisson limit theorem for a strongly ergodic non. Markov chains and hidden markov models by nick whiteley and anthony lee university of bristol and university of warwick we obtain a perfect sampling characterization of weak ergodicity for backward products of nite stochastic matrices, and equivalently, simultaneous tail triviality of the corresponding nonhomogeneous markov chains. Therefore, most publications concern this situation only see, e. Finitestate markov chains furthermore, prx n j x n. A nonstationary markov chain is called weakly ergodic if for all. Nonhomogeneous markov chains and their applications. This dissertation establishes new results in the characterization of strong ergodicity for continuous time markov chains. The triangular array of inhomogeneous markov chains fx n. Markov setchains as abstractions of stochastic hybrid.
He used, for the first time, martingales in the study of nonhomogeneous chains and decomposed the set of states into classes in a manner which was shown later to be the tail afield decomposition see cohn 1974b. A geometric approach to ergodic nonhomogeneous markov. Pdf weak ergodicity of nonhomogeneous markov chains on. Inspired by the work of daubechies and lagarias on a set of matrices with convergent infinite products, we study the geometric approach to the classical problem of weakly ergodic non homogeneous markov chains. Further in section 4, we find necessary and sufficient conditions for the weak ergodicity of nonhomogeneous discrete markov chains ndmc, which. Markov chains are an essential component of markov chain monte carlo mcmc techniques. Weak ergodicity in nonhomogeneous markov chains volume 54 issue 2 j. A main objective is to get convergence properties as well as rate of convergence of stochastic algorithms based on general markov chains as, for instance, in markov search for opti. The existing key inequalities related to the hajnal inequality in the literature are. On the weak ergodicity of nonhomogeneous markov chains. This study studies the consensus of multiagent systems on timevarying network topologies. All textbooks and lecture notes i could find initially introduce markov chains this way but then quickly restrict themselves to the timehomogeneous case where you have one transition matrix. Ergodicity in parametric non stationary markov chains. Occupancy distributions in markov chains via doeblins.
Fischer, lynch, paterson asynchronous impossibility result. A very mild sufficient condition of the consensus reaching that allows the communication among agents to be timedependent and directed is obtained by estimating the dobrushin coefficient of ergodicity. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered banach space with a base obsb, and study its properties. The determinant of a long enough word has a value close to zero, so that the rows of the word are almost identical. Some lemmas for nonhomogeneous markov chains a nonhomogeneous markov chain is called strongly ergodic 4 if there exists a matrix. Farrukh mukhamedov, ahmed alrawashdeh submitted on 17 jan 2020. Simple examples of the use of nash inequalities for finite markov chains. Weak ergodicity of nonhomogeneous markov chains on noncommutative l1 spaces. This book is a modern presentation of the semimartingale or lyapunov function method applied to nearcritical stochastic systems, exemplified by nonhomogeneous random walks. Characterizations of strong ergodicity for continuous time. To get a better understanding of what a markov chain is, and further, how it can be used to sample form a distribution, this post introduces and applies a few basic concepts. Weak ergodicity as defined in definition 1 in general does not imply strong ergodicity. Dec 01, 2008 weak ergodicity as defined in definition 1 in general does not imply strong ergodicity.
We give some basic definitions of the vector norms and matrix norms that will be used here. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered banach spaces with a base obsb, and study its properties. This may occur in nonhomogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases. Sep 22, 2015 it is known that the dobrushins ergodicity coefficient is one of the effective tools to study the behavior of nonhomogeneous markov chains. Independence gchange can be good iacademia lets you. We are interested in the class of stochastic processes based on nonhomogeneous markov chains. It is known that the dobrushins ergodicity coefficient is one of the effective tools to study a behavior of nonhomogeneous markov chains. Stability and approximation of invariant measures of. Monographs on statistics and applied probability 80 365400. The norm is used to give the following definitions of weak and strong ergodicity. It is known that the dobrushins ergodicity coefficient is one of the effective tools to study a behavior of non homogeneous markov chains. The proof uses the hilbert projection metric and the fact that the linear cocycle generated by the markov chain is a uniformly contractive mapping of the positive cone. In his first paper on markov chains, published in 1906, markov showed that under certain conditions the average outcomes of the markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold.
1157 820 1062 1569 1471 1209 1555 573 941 467 569 297 676 248 835 377 697 999 1140 531 69 1364 852 384 513 308 1424 1343