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.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

On sample path properties of semistable processes

This paper contains three main results: In the first result a correspondence principle between semistable measures on L_p, 1 ≤ p < ∞, and Banach space valued semistable processes is established. In the second result it is shown that the paths of a Banach space valued semistable process belong to L_p with probability zero or one, and necessary and sufficient conditions for the two alternatives to hold are given. In the third result necessary and sufficient conditions are given for almost sure path absolute continuity for certain Banach space valued semistable processes.     

Quasi-stationarity and quasi-ergodicity of general Markov processes

In this paper, we study the quasi-stationarity and quasi-ergodicity of general Markov processes. We show, among other things, that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t, x, y) (with respect to m) which is bounded in (x, y) for every t > 0, then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution. We also present several classes of Markov processes satisfying the above conditions.     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.     

On stationary Markov processes with polynomial conditional moments

We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say m is a polynomial of degree at most m. We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence {(α_n, p_n)}_{n ? 0} where α′_ns are some positive reals while p′_ns are some monic orthogonal polynomials. Bakry and Mazet (Séminaire de Probabilit?s, vol. 37, 2003) showed that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression.

We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein–Uhlenbeck (OU) processes or can also be understood as time scale transformations of Lévy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of Lévy processes then they are not stationary unless we deal with classical OU process. Conversely, time scale transformations of stationary processes with independent regression property are not Lévy unless we deal with classical OU process.     

Extending the Wong-Zakai theorem to reversible Markov processes

We show how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak H?lder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We further prove that certain polygonal approximations to these paths and their areas converge in p-variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant generalization of the classical result of Wong-Zakai on the approximation of solutions to stochastic differential equations. Our results allow us to construct solutions to differential equations driven by reversible Markov processes of finite p-variation with p<4. Received May 18, 2001 / final version received April 3, 2001?Published online April 8, 2002     

On performance comparison of MR/GI/1 queues

In this paper we are interested in the effect that dependencies in the arrival process to a queue have on queueing properties such as mean queue length and mean waiting time. We start with a review of the well known relations used to compare random variables and random vectors, e.g., stochastic orderings, stochastic increasing convexity, and strong stochastic increasing concavity. These relations and others are used to compare interarrival times in Markov renewal processes first in the case where the interarrival time distributions depend only on the current state in the underlying Markov chain and then in the general case where these interarrivai times depend on both the current state and the next state in that chain. These results are used to study a problem previously considered by Patuwo et al. [14].Then, in order to keep the marginal distributions of the interarrivai times constant, we build a particular transition matrix for the underlying Markov chain depending on a single parameter,p. This Markov renewal process is used in the Patuwo et al. [14] problem so as to investigate the behavior of the mean queue length and mean waiting time on a correlation measure depending only onp. As constructed, the interarrival time distributions do not depend onp so that the effects we find depend only on correlation in the arrival process.As a result of this latter construction, we find that the mean queue length is always larger in the case where correlations are non-zero than they are in the more usual case of renewal arrivals (i.e., where the correlations are zero). The implications of our results are clear.     

On Finite Markov Chain Imbedding and Its Applications

The finite Markov Chain Imbedding technique has been successfully applied in various fields for finding the exact or approximate distributions of runs and patterns under independent and identically distributed or Markov dependent trials. In this paper, we derive a new recursive equation for distribution of scan statistic using the finite Markov chain imbedding technique. We also address the problem of obtaining transition probabilities of the imbedded Markov chain by introducing a notion termed Double Finite Markov Chain Imbedding where transition probabilities are obtained by using the finite Markov chain imbedding technique again. Applications for random permutation model in chemistry and coupon collector’s problem are given to illustrate our idea.     

Rational Brauer characters

It is known that every finite group of even order has a non-trivial complex irreducible character which is rational valued. We prove the modular version of this result: If p is an odd prime and G is any finite group of even order, then G has a non-trivial irreducible p-Brauer character which is rational valued. The first author is partially supported by the Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01, while the second gratefully acknowledges the support of the NSA (grant H98230-04-0066).     

Non-equilibrium entropy on stationary Markov processes

We study the evolution of probability measures under the action of stationary Markov processes by means of a non-equilibrium entropy defined in terms of a convex function . We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of for a wide class of convex functions, including ₀(t)=t log t. We also prove that this is equivalent to the convergence of all the densities of a finite norm to a uniform density, on the Orlicz spaces related to , which include the L ^p -spaces for p>1. By means of the quadratic function ₂(t)=t ²–1, we relate the non-equlibrium entropies defined by the past -algebras of a K-dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.Partially supported by DIB Universidad de Chile, E19468412.     

Upper functions for lévy processes having only negative jumps

Let X_t,t ≥ 0 be a real valued process with stationary independentincrements having only negative jumps. We obtain b(t) such that lim sup X_t )/b(t) equals a finite positive constant with probability one as t → 0 and t → ∞ under extra condition. The hypotheses about the behavior of Lévy measure near zero and infinity are necessary to guarantee that the lim sup is positive     

Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

We study the fractal properties of distributions of random variables digits of polybasic Q-representations (a generalization of n-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.     

The Symmetric Genus of p-Groups

Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has symmetric genus not congruent to 1(mod p ³) if and only if G is in one of 10 families of groups. The genus formula for each of these 10 families of groups is determined. A consequence of this classification is that almost all positive integers that are the genus of a p-group are congruent to 1(mod p ³). Finally, the integers that occur as the symmetric genus of a p-group with Frattini-class 2 have density zero in the positive integers.     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

Limit theorems for <Emphasis Type="Italic">p</Emphasis>-adic valued asymmetric semistable laws and processes

Limit distributions of scaled sums of p-adic valued i.i.d. are characterized as semistable laws, and a condition to assure the weak convergence of a scaled sum is verified. The limit supremum of the norm of the weakly convergent scaled sum is divergent in fact, and the exact growth rate of the sum is given. It is also shown that, if a scaled sum including a time parameter in the number of the added i.i.d. is considered, the semigroup of the limit distributions corresponds to a p-adic valued Markov process having right continuous sample paths with left limits. These are generalizations of the former results for rotation-symmetric i.i.d., with some necessary modifications.     

The density of the strong symmetric genus values of p-groups

Let G be a finite group. The strong symmetric genus σ⁰(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Let p a prime, and let J_p be the set of integers g for which there is a p-group of strong symmetric genus g. We show that the set J_p has density zero in the set of positive integers.     

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.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.