Completeness is determined by any non-algebraic trajectory

It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.     

Uniform ergodic convergence and averaging along Markov chain trajectories

LetP(x, A) be a transition probability on a measurable space (S, Σ) and letX _n be the associated Markov chain.Theorem. Letf∈B(S, Σ). Then for anyx∈S we haveP _x a.s. $$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {f(X_k ) \geqslant } \mathop {\underline {\lim } }\limits_{n \to \infty } \mathop {\inf }\limits_{x \in S} \frac{1}{n}\sum\limits_{k = 1}^n {P^k f(x)} $$ and (implied by it) a corresponding inequality for the lim. If 1/n∑ _k=1 ⁿ P ^k f converges uniformly, then for everyx∈S, 1/n ∑ _k=1 ⁿ f(X _k ) convergesP _x a.s. Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n∑ _k=1 ⁿ μ ^k *f and of 1/n∑ _k=1 ⁿ f(X _k ) via that ofΨ _n *f(x)=m(A _n )^?1∫ _An f(xt), where {A _n } is a Følner sequence, in the following cases: (i)f is left uniformly continuous (ii) μ is spread out (iii)G is Abelian. Non-Abelian Example: Let μ be adapted and spread-out on a nilpotent σ-compact locally compact groupG, and let {A _n } be a Følner sequence. If forf∈B(G, ∑) m(A _n )^?1∫ _An f(xt)dm(t) converges uniformly, then 1/n∑ _k=1 ⁿ f(X _k ) converges uniformly, andP _x convergesP _x a.s. for everyx∈G.     

Sensitivity of the stationary distribution vector for an ergodic Markov chain

Stationary distribution vectors p^∞ for Markov chains with associated transition matrices T are important in the analysis of many models in the mathematical sciences, such as queuing networks, input-output economic models, and compartmental tracer analysis models. The purpose of this paper is to provide insight into the sensitivity of p^∞ to perturbations in the transition probabilities of T and to understand some of the difficulties in computing an accurate p^∞. The group inverse A^# of I − T is shown to be of fundamental importance in understanding sensitivity or conditioning of p^∞. The main result shows that if there is a state that is accessible from every other state and the corresponding column of T has no small off-diagonal elements, then p^∞ cannot be sensitive to small perturbations in T. Ecological examples are given. A new algorithm for calculating A^# is described.     

A Poisson limit theorem for a strongly ergodic non-homogeneous Markov chain

A strongly ergodic non-homogeneous Markov chain is considered in the paper. As an analog of the Poisson limit theorem for a homogeneous Markov chain recurring to small cylindrical sets, a Poisson limit theorem is given for the non-homogeneous Markov chain. Meanwhile, some interesting results about approximation independence and probabilities of small cylindrical sets are given.     

On asymptotics of the potential of a countable ergodic Markov chain

For a class of functionsf, the convergence in Abel's sense is proved for the potential _no P ⁿ f(i) of a uniform ergodic Markov chain in a countable phase space. Several corollaries are obtained which are useful from the point of view of the possible application to CLT (the central limit theorem) for Markov chains. In particular, we establish the condition equivalent to the boundedness of the second moment for the time of the first return into the state.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 2, pp. 265–269, February, 1993.     

Some results associated with the longest run in a strongly ergodic Markov chain

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain.Let {Xa} a∈Z + be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1,2,...}.Let TS be an arbitrary finite subset of S.Denote by Ln the length of the longest run of consecutive i's for i∈T,that occurs in the sequence X1,...,Xn.In this paper,we obtain a limit law and a week version of an Erds-Rényi type law for Ln.A large deviation result of Ln is also discussed.     

Markov Chains with Positive Transitions Are Not Determined by Any p -Marginals

 We prove that if is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number p, there exists a continuum of finite valued non Markovian processes which have the p-marginal distributions of X and with positive entropy, whereas for an irrational rotation and essentially bounded real measurable function f with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain.     

An embedded Markov chain approach to stock rationing

We propose a new method for the analysis of lot-per-lot inventory systems with backorders under rationing. We introduce an embedded Markov chain that approximates the state-transition probabilities. We provide a recursive procedure for generating these probabilities and obtain the steady-state distribution.     

An optimal pricing policy under a Markov chain model

This paper is about an optimal pricing control under a Markov chain model. The objective is to dynamically adjust the product price over time to maximize a discounted reward function. It is shown that the optimal control policy is of threshold type. Closed-form solutions are obtained. A numerical example is also provided to illustrate our results.

     

Some chain geometries determined by transformation groups

In the paper we propose a modification of the classical construction of the (Minkowskian) incidence structures based on permutation groups. Dropping out explicit assumptions concerning rigidity and transitivity (and assuming an arbitrary finite ”dimension”) we obtain a wider class of structures. Their geometrical properties are studied; in particular, we establish their automorphism groups and discuss some problems related to axiomatic characterization.     

Adic realizations of ergodic actions by homeomorphisms of Markov compacta and ordered Bratteli diagrams

For any ergodic transformation T of a Lebesgue space (X, μ), it is possible to introduce a topology τ on X such that (a) X becomes a totally disconnected compactum (a Cantor set) with a Markov structure, and μ becomes a Borel Markov measure; (b) T becomes a minimal strictly ergodic homeomorphism of (X, τ); (c) the orbit partition of T is the tail partition of the Markov compactum (up to two classes of the partition). The Markov compactum structure is the same as the path structure of the Bratteli diagram for some AF-algebra. Bibliography: 19 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 120–126.     

An interacting multiple model algorithm with a switching Markov chain

A Markov chain plays an important role in an interacting multiple model (IMM) algorithm which has been shown to be effective for target tracking systems. Such systems are described by a mixing of continuous states and discrete modes. The switching between system modes is governed by a Markov chain. In real world applications, this Markov chain may change or needs to be changed. Therefore, one may be concerned about a target tracking algorithm with the switching of a Markov chain. This paper concentrates on fault-tolerant algorithm design and algorithm analysis of IMM estimation with the switching of a Markov chain. Monte Carlo simulations are carried out and several conclusions are given.     

Parallel Markov chain Monte Carlo Simulation by Pre-Fetching

In recent years, parallel processing has become widely available to researchers. It can be applied in an obvious way in the context of Monte Carlo simulation, but techniques for “parallelizing” Markov chain Monte Carlo (MCMC) algorithms are not so obvious, apart from the natural approach of generating multiple chains in parallel. Although generation of parallel chains is generally the easiest approach, in cases where burn-in is a serious problem, it is often desirable to use parallelization to speed up generation of a single chain. This article briefly discusses some existing methods for parallelization of MCMC algorithms, and proposes a new “pre-fetching” algorithm to parallelize generation of a single chain.     

The lollipop graph is determined by its Q-spectrum

A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q, any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by H_n,p, is obtained by appending a cycle C_p to a pendant vertex of a path P_n−p. In this paper, it is proved that all lollipop graphs are determined by their Q-spectra.