scholarly journals Yaglom limits can depend on the starting state

2018 ◽  
Vol 50 (01) ◽  
pp. 1-34
Author(s):  
R. D. Foley ◽  
D. R. McDonald
Keyword(s):  
Markov Chain ◽  
Transition Matrix ◽  
Martin Boundary ◽  
Countable State Space ◽  
Convergence Parameter ◽  
Starting State ◽  
Yaglom Limit ◽  
Countable State ◽  
Transient Markov Chain ◽  
Quasi Stationary Distribution

AbstractWe construct a simple example, surely known to Harry Kesten, of anR-transient Markov chain on a countable state spaceS∪ {δ}, where δ is absorbing. The transition matrixKonSis irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given nonabsorption. Each starting statex∈Sresults in a different Yaglom limit. Each Yaglom limit is anR-1-invariant quasi-stationary distribution, whereRis the convergence parameter ofK. Yaglom limits that depend on the starting state are related to a nontrivialR-1-Martin boundary.

scholarly journals ON THE STABILITY OF A BANDWIDTH PACKING ALGORITHM

2000 ◽  
Vol 14 (1) ◽  
pp. 57-79 ◽  
Author(s):  
Jean-François Dantzer ◽  
Mostafa Haddani ◽  
Philippe Robert
Keyword(s):  
Markov Chain ◽  
Bandwidth Allocation ◽  
Transient Behavior ◽  
Finite Markov Chain ◽  
Countable State Space ◽  
Countable State ◽  
The Stability ◽  
Bandwidth Packing ◽  
Bandwidth Allocation Algorithm ◽  
Transient Markov Chain

The stability properties of the bandwidth allocation algorithm First Fit are analyzed for some distributions on the sizes of the requests. Fluid limits are used to get the ergodicity results. When there are two possible sizes, the description of the transient behavior involves a finite Markov chain on the exit states of a transient Markov chain on a countable state space. The explicit expression of this exit matrix is given.

scholarly journals Markov chains with exponential return times are finitary

2020 ◽  
pp. 1-9
Author(s):  
OMER ANGEL ◽  
YINON SPINKA
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Proper Subgroup ◽  
Ergodic Markov Chain ◽  
Countable State Space ◽  
Return Times ◽  
Exponential Tails ◽  
Stationary Version ◽  
Countable State

Abstract Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities

1973 ◽  
Vol 73 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Gerald S. Goodman ◽  
S. Johansen
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Continuous Transition ◽  
Countable State Space ◽  
Countable State ◽  
Uniformly Continuous

1. SummaryWe shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

1989 ◽  
Vol 26 (3) ◽  
pp. 643-648 ◽  
Author(s):  
A. I. Zeifman
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

A Lyapunov Criterion for Invariant Probabilities with Geometric Tail

1998 ◽  
Vol 12 (3) ◽  
pp. 387-391
Author(s):  
Jean B. Lasserre
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Sufficient Condition ◽  
Countable State Space ◽  
Countable State

Given a Markov chain on a countable state space, we present a Lyapunov (sufficient) condition for existence of an invariant probability with a geometric tail.

An infinite particle system with continual input and random death

1978 ◽  
Vol 10 (04) ◽  
pp. 764-787
Author(s):  
J. N. McDonald ◽  
N. A. Weiss
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Limit Theorems ◽  
Particle System ◽  
Countable State Space ◽  
Infinite Particle System ◽  
Large Numbers ◽  
Countable State ◽  
Infinite Particle ◽  
Number Of Particles

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

On the existence of quasi-stationary distributions in denumerable R-transient Markov chains

1992 ◽  
Vol 29 (01) ◽  
pp. 21-36 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
State Space ◽  
Stationary Distribution ◽  
Practical Importance ◽  
Prime Concern ◽  
Continuous Random Walk ◽  
Transient Markov Chain ◽  
Quasi Stationary Distribution

Let {Xn, n= 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩+= {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix𝐏= (pij), and letpij(n) =P∈Xn=j, Xk∈ 𝒩+fork= 0, 1, ···,n|X0=i],i, j𝒩+. The prime concern of this paper is conditions for the existence of the limits,qijsay, ofasn →∞. Ifthe distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vectorx= (xi) satisfyingrxT=xT𝐏andexists, whereris the convergence norm of𝐏, i.e.r=R–1andand T denotes transpose, then it is unique, positive elementwise, andqij(n) necessarily converge toxjasn →∞.Unlike existing results in the literature, our results can be applied even to theR-null andR-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix isR-transient is discussed to demonstrate the usefulness of our results.

One-dimensional branching random walks in a Markovian random environment

2000 ◽  
Vol 37 (4) ◽  
pp. 1157-1163 ◽  
Author(s):  
F. P. Machado ◽  
S. Yu. Popov
Keyword(s):  
Markov Chain ◽  
Random Environment ◽  
Branching Random Walk ◽  
Countable State Space ◽  
One Dimensional ◽  
Branching Random Walks ◽  
Countable State ◽  
Markovian Random Environment ◽  
Recurrent Markov Chain ◽  
Step Transition

We study a one-dimensional supercritical branching random walk in a non-i.i.d. random environment, which considers both the branching mechanism and the step transition. This random environment is constructed using a recurrent Markov chain on a finite or countable state space. Criteria of (strong) recurrence and transience are presented for this model.

Markov renewal theory

1969 ◽  
Vol 1 (02) ◽  
pp. 123-187 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
State Space ◽  
Renewal Theory ◽  
The State ◽  
Countable State Space ◽  
Countable State ◽  
Markov Renewal Theory ◽  
Random Amount

Consider a stochastic process X(t) (t ≧ 0) taking values in a countable state space, say, {1, 2,3, …}. To be picturesque we think of X(t) as the state which a particle is in at epoch t. Suppose the particle moves from state to state in such a way that the successive states visited form a Markov chain, and that the particle stays in a given state a random amount of time depending on the state it is in as well as on the state to be visited next. Below is a possible realization of such a process.

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