Existence of Gibbs measures for countable Markov shifts

We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbanski (2001) who showed that this condition is sufficient.

     

Singular stationary measures are not always fractal

We know of few explicit results to insure that stationary measures are simultaneously (i) singular, (ii) nonatomic, (iii) with interval support, and (iv) unique. Such results would appear useful, to further separate the analytic notion of singular from the geometric notion of fractal. We prove two general theorems, one for maps of [0,1] into [0, 1], the other for 2×2 random matrices. In each setting, we study measures supported on two points of the transformation space, and we provide sufficient conditions to insure that the stationary measures satisfy (i)–(iv).     

Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

     

Analysis of convergence rates of some Gibbs samplers on continuous state spaces

We use a non-Markovian coupling and small modifications of techniques from the theory of finite Markov chains to analyze some Markov chains on continuous state spaces. The first is a generalization of a sampler introduced by Randall and Winkler, and the second a Gibbs sampler on narrow contingency tables.     

Strong stationary duality for continuous-time Markov chains. Part I: Theory

LetX(t), 0t<, be an ergodic continuous-time Markov chain with finite or countably infinite state space. We construct astrong stationary dual chainX ^* whose first hitting times yield bounds on the convergence to stationarity forX. The development follows closely the discrete-time theory of Diaconis and Fill.^(2,3) However, for applicability it is important that we formulate our results in terms of infinitesimal rates, and this raises new issues.     

On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues ν _j of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates ν _j . We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper’s final section we present extensions of our results to more general chains. Research supported by NSF grant DMS–0406104, and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Gibbs Measures and Markov Random Fields with Association $$I $$

We introduce the notions of a Gibbs measure with the corresponding potential with association (where is a subset of the set ) of a Markov random field with memory and measure with association . It is proved that these three notions are equivalent.     

Ergodicity of canonical Gibbs measures with respect to the diffeomorphism group

For general potentials we prove that every canonical Gibbs measure on configurations over a manifold X is quasi‐invariant w.r.t. the group of diffeomorphisms on X. We show that this quasi‐invariance property also characterizes the class of canonical Gibbs measures. From this we conclude that the extremal canonical Gibbs measures are just the ergodic ones w.r.t. the diffeomorphism group. Thus we provide a whole class of different irreducible representations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Additive Properties of the Dufresne Laws and Their Multivariate Extension

The Dufresne laws are defined on the positive line by their Mellin transform , where the a _i and b _j are positive numbers, with pq, and where (x) _s denotes (x+s)/(x). Typical examples are the laws of products of independent random variables with gamma and beta distributions. They occur as the stationary distribution of certain Markov chains (X _n) on defined by where X ₀, (A ₁, B ₁),..., (A _n, B _n),... are independent. This paper gives some explicit examples of such Markov chains. One of them is surprisingly related to the golden number. While the properties of the product of two independent Dufresne random variables are trivial, we give several properties of their sum: the hypergeometric functions are the main tool here. The paper ends with an extension of these Dufresne laws to the space of positive definite matrices and to symmetric cones.     

随机环境中马氏链与马氏双链间的相互关系

比较圆满地解决了单无限和双无限环境及其对应的双链和原过程四者间的相互关系．特别地,澄清了一些误解,纠正了其中的错误结论,为进一步深入研究随机环境中马氏链提供了非常明晰的基本概念．     

Existence and uniqueness of an invariant probability for a class of Feller Markov chains

We consider the class of Feller Markov chains on a phase spaceX whose kernels mapC ₀ (X), the space of bounded continuous functions that vanish at infinity, into itself. We provide a necessaryand sufficient condition for the existence of an invariant probability measure using a generalized Farkas Lemma. This condition is a Lyapunov type criterion that can be checked in practice. We also provide a necessaryand sufficient condition for existence of aunique invariant probability measure. When the spaceX is compact, the conditions simplify.     

Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

Let {X_n} be a ?-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {X_n} the conditions for ergodicity will be met if there is a compact set K and an ? > 0 such that $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 — 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? ??}$ whenever x lies outside K and $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ∣ X}{}_{\mathrm{n}}\text{=x}}$ is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ? 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? 0}$ for all x outside K. An application to queueing theory is given.     

Convergence of a transition probability tensor of a higher‐order Markov chain to the stationary probability vector

In this paper, first we introduce a new tensor product for a transition probability tensor originating from a higher‐order Markov chain. Subsequently, some properties of the new tensor product are explained, and its relationship with the stationary probability vector is studied. Also, similarity between results obtained by this new product and the first‐order case is shown. Furthermore, we prove the convergence of a transition probability tensor to the stationary probability vector. Finally, we show how to achieve a stationary probability vector with some numerical examples and make some comparison between the proposed method and another existing method for obtaining stationary probability vectors. Copyright © 2016 John Wiley & Sons, Ltd.     

The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof

A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray–Knight theorem. Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time T for an ergodic birth-and-death chain on {0,…,d} in continuous time with generator G, started in state 0, is distributed as a sum of d independent exponential random variables whose rate parameters are the nonzero eigenvalues of −G. Our approach yields the first (sample-path) construction of such a T for which individual such exponentials summing to T can be explicitly identified. Research of J.A. Fill was supported by NSF grant DMS–0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

A central limit theorem for mixing stationary point processes

Suppose A₀ is a strictly stationary, second order point process on Z^d that is ?-mixing. The particles initially present are then continually subjected to random translations via random walks. If A_n is the point process resulting at time n, then we prove, under certain technical conditions, that the total occupation time by time n of a finite nonempty subset B of Z^d, namely, S_n(B)=Σⁿ_k=1A_k(B), is asymptotically normally distributed.     

Generalized differentiable product measures

A class of measures on ℝ^∞ determined by sequences of functions of finitely many variables is considered. An existence theorem for such measures is proved, and their properties are examined. Examples are presented. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 37–55, January, 1998. The author is greatly indebted to O. V. Zimina for many stimulating discussions. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01 01701.     

Remarks on Estimation Problem for Stationary Processes in Continuous Time

The problem of estimating of the law (in the space of the paths) and the common marginal distribution for a strictly stationary ergodic process X is discussed. We show, in particular, that:(1) The empirical measure

Restoration of Gibbsianness for projected and FKG renormalized measures

We restore part of the thermodynamic formalism for some renormalized measures that are known to be non-Gibbsian. We determine a necessary and sufficient condition for consistency with a specification that is quasilocal only in a fixed direction. This condition is then applied to models with FKG monotonicity and to models with appropriate directional continuity rates, in particular to (noisy) decimations or projections of the Ising model. In this way we establish: (i) the validity of the second part of the variational principle for projected and FKG block-renormalized measures, and (ii) the almost quasilocality of FKG block-renormalized + and – measures.     

Chutes and Ladders in Markov Chains

We investigate how the stationary distribution of a Markov chain changes when transitions from a single state are modified. In particular, adding a single directed edge to nearest neighbor random walk on a finite discrete torus in dimensions one, two, or three changes the stationary distribution linearly, logarithmically, or only locally. Related results are derived for birth and death chains approximating Bessel diffusions and for random walk on the Sierpinski gasket.