Markov chains with exponential return times are finitary

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.

Regularity of Geometric quadratic stochastic operator generated by 2-partition of infinite points

In this research, we construct a class of quadratic stochastic operator called Geometric quadratic stochastic operator generated by arbitrary 2-partition of infinite points on a countable state space , where . We also study the limiting behavior of such operator by proving the existence of the limit of the sequence through the convergence of the trajectory to a unique fixed point. It is established that such operator is a regular transformation.

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.

Yaglom limits can depend on the starting state

We 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.