Optimality in transient markov chains and linear programming

It is shown how a discrete Markov programming problem can be transformed, using a linear program, into an equivalent problem from which the optimal decision rule can be trivially deduced. This transformation is applied to problems which have either transient probabilities or discounted costs.This research was supported by the National Research Council of Canada, Grant A7751.     

On multiple Markov chains

The aim of this paper is to examine multiple Markov dependence for the discrete as well as for the continuous parameter case. In both cases the Markov property with arbitrary parameter values is investigated and it is shown that it leads to the degeneration of the multiple Markov dependence to the simple one.     

Nash inequalities for finite Markov chains

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) steps are necessary and suffice to reach stationarity. We consider local Poincaré inequalities and use them to prove Nash inequalities. These are bounds onl ₂-norms in terms of Dirichlet forms andl ₁-norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.     

On the representation of finite multiple Markov chains by weighted circuits

In this paper circuit chains of superior order are defined as multiple Markov chains for which transition probabilities are expressed in terms of the weights of a finite class of circuits in a finite set, in connection with kinetic properties along the circuits. Conversely, it is proved that if we join any finite doubly infinite strictly stationary Markov chain of order r for which transitions hold cyclically with a second chain with the same transitions for the inverse time-sense, then they may be represented as circuit chains of order r.     

Eigentime identity for asymmetric finite Markov chains

Two kinds of eigentime identity for asymmetric finite Markov chains are proved both in the ergodic case and the transient case.     

On additive functionals of Markov chains

Let{S _n } _n=0 ^∞ be a Harris-recurrent Markov chain on a measurable state space. We prove strong approximation results for the additive functionals . Research supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. Research supported by an NSERC Canada Grant, Carleton University. Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6.     

On perturbation bounds for continuous-time Markov chains

We suggest an approach to obtaining general estimates of stability in terms of special “weighted” norms related to total variation. Two important classes of continuous-time Markov chains are considered for which it is possible to obtain exact convergence rate estimates (and hence, guarantee exact stability estimates): birth–death–catastrophes processes, and queueing models with batch arrivals and group services.     

On finite long run costs and rewards in infinite Markov chains

Conditions for the finiteness of long run costs and rewards associated with infinite recurrent Markov chains that may be discrete or continuous in time are considered. Without resorting to results from the theory of Markov processes on general state spaces we provide instructive proofs in the course of which we derive auxiliary results that are of interest in themselves. Potential applications of the finiteness conditions are outlined in order to elucidate their high practical relevance.     

Gradient flows of the entropy for finite Markov chains

Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in Rn by Jordan, Kinderlehrer and Otto (1998). The metric W is similar to, but different from, the L²-Wasserstein metric, and is defined via a discrete variant of the Benamou-Brenier formula.     

Denumerable controlled Markov chains with average reward criterion: Sample path optimality

We consider discrete-time nonlinear controlled stochastic systems, modeled by controlled Makov chains with denumerable state space and compact action space. The corresponding stochastic control problem of maximizing average rewards in the long-run is studied. Departing from the most common position which usesexpected values of rewards, we focus on a sample path analysis of the stream of states/rewards. Under a Lyapunov function condition, we show that stationary policies obtained from the average reward optimality equation are not only average reward optimal, but indeed sample path average reward optimal, for almost all sample paths.Research supported by a U.S.-México Collaborative Research Program funded by the National Science Foundation under grant NSF-INT 9201430, and by CONACyT-MEXICO.Partially supported by the MAXTOR Foundation for applied Probability and Statistics, under grant No. 01-01-56/04-93.Research partially supported by the Engineering Foundation under grant RI-A-93-10, and by a grant from the AT&T Foundation.     

关于DHMM转化为齐次马尔可夫链的一个定理及其证明

本给出一个将DHMM转化为齐次马尔可夫链的定理，该定理提供了利用在理论上比较完善的齐次马尔可夫链来研究DHMM的一个方法．     

On a result for finite Markov chains

In an undergraduate course on stochastic processes, Markov chains are discussed in great detail. Textbooks on stochastic processes provide interesting properties of finite Markov chains. This note discusses one such property regarding the number of steps in which a state is reachable or accessible from another state in a finite Markov chain with M (≥ 2) states.     

On mixtures of distributions of Markov chains

Let X be a chain with discrete state space I, and V be the matrix of entries V_i,n, where V_i,n denotes the position of the process immediately after the nth visit to i. We prove that the law of X is a mixture of laws of Markov chains if and only if the distribution of V is invariant under finite permutations within rows (i.e., the V_i,n's are partially exchangeable in the sense of de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of V and Markov exchangeability in the sense of Diaconis and Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the Diaconis and Freedman basic result for mixtures of Markov chains.     

A different construction of Gaussian fields from Markov chains: Dirichlet covariances

We study a class of Gaussian random fields with negative correlations. These fields are easy to simulate. They are defined in a natural way from a Markov chain that has the index space of the Gaussian field as its state space. In parallel with Dynkin's investigation of Gaussian fields having covariance given by the Green's function of a Markov process, we develop connections between the occupation times of the Markov chain and the prediction properties of the Gaussian field. Our interest in such fields was initiated by their appearance in random matrix theory.     

On an extremal property of Markov chains and sufficiency of Markov strategies in Markov decision processes with the Dubins-Savage criterion

An inequality regarding the minimum ofP(lim inf(X _n D _n )) is proved for a class of random sequences. This result is related to the problem of sufficiency of Markov strategies for Markov decision processes with the Dubins-Savage criterion, the asymptotical behaviour of nonhomogeneous Markov chains, and some other problems.     

On the structure of 1-dependent Markov chains

Any stationary 1-dependent Markov chain with up to four states is a 2-block factor of independent, identically distributed random variables. There is a stationary 1-dependent Markov chain with five states which is not, even though every 1-dependent renewal process is a 2-block factor.     

On Markov chains induced by partitioned transition probability matrices

Let S be a denumerable state space and let P be a transition probability matrix on S. If a denumerable set M of nonnegative matrices is such that the sum of the matrices is equal to P, then we call M a partition of P.     

On the range of recurrent Markov chains

Let {X_n}₀^∞ be an irreducible recurrent Markov Chain on the nonnegative integers. A result of Chosid and Isaac (1978) gives a sufficient condition for n^?1R_n → 0 w.p.1. where R_n is the range of the chain. We give an alternative proof using Kingman's subadditive ergodic theorem (Kingman, 1973). Some examples are also given.     

Spectral analysis of finite Markov chains with spherical symmetries

We generalize the classical Fourier analysis of Gelfand pairs to the setting of groups acting not transitively on a set X. We use this analysis to determine the spectrum of several random walks on graphs. Moreover, as byproduct, we show that, for a new urn diffusion model, the cut-off phenomenon holds.