Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compressibility and Stochastic Stability of Monotone Markov Chain

Sergey Foss, Michael Scheutzow

Abstract

For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular, we generalise a basic result by Bhattacharya and Majumdar (2007) where a certain form of mixing, or swap condition was assumed uniformly over the state space. We do not rely on continuity properties of transition probabilities.

Compressibility and Stochastic Stability of Monotone Markov Chain

Abstract

For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular, we generalise a basic result by Bhattacharya and Majumdar (2007) where a certain form of mixing, or swap condition was assumed uniformly over the state space. We do not rely on continuity properties of transition probabilities.
Paper Structure (12 sections, 15 theorems, 39 equations)

This paper contains 12 sections, 15 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Monotone Markov chains
  3. Sample-path representations
  4. Stochastic monotonicity
  5. Conditions for compressibility
  6. Generic results
  7. Sufficient conditions for achievability of the set $M$
  8. Conditions for existence of a stationary regime
  9. Conditions for uniqueness of the stationary regime
  10. Stochastic stability
  11. A Markov chain as a stochastic recursion
  12. Generalisations

Key Result

Proposition 2.1

(Strassen's theorem, see KKO or Li). Assume that $\cal E$ is an ordered Polish space. If $\mu_1,\,\mu_2 \in {\cal M}_1(\cal E)$ satisfy $\mu_1 \preceq \mu_2$ then there is a coupling $\lambda$ of $\mu_1$ and $\mu_2$ for which $\lambda(M)=1$.

Theorems & Definitions (43)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • Lemma 2.8
  • ...and 33 more