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Strict irreducibility of Markov chains and ergodicity of skew products

Pablo Lummerzheim, Felix Pogorzelski, Elias Zimmermann

Abstract

We consider a family of measure preserving transformations, which act on a common probability space and are chosen at random by a stationary ergodic Markov chain. This setting defines an instance of a random dynamical system (RDS), which may be described in terms of a step skew product. In many contexts it is desirable to know whether ergodicity of the family implies ergodicity of the skew product. Introducing the notion of strict irreducibility for Markov kernels we shall characterize the class of Markov chains for which the aforementioned implication holds true. We thereby extend a sufficient condition of Bufetov for the case of finite state Markov chains to general state spaces and show that it is in fact also necessary. As an application we obtain an explicit description of the limit in ergodic theorems for a suitable class of random transformations.

Strict irreducibility of Markov chains and ergodicity of skew products

Abstract

We consider a family of measure preserving transformations, which act on a common probability space and are chosen at random by a stationary ergodic Markov chain. This setting defines an instance of a random dynamical system (RDS), which may be described in terms of a step skew product. In many contexts it is desirable to know whether ergodicity of the family implies ergodicity of the skew product. Introducing the notion of strict irreducibility for Markov kernels we shall characterize the class of Markov chains for which the aforementioned implication holds true. We thereby extend a sufficient condition of Bufetov for the case of finite state Markov chains to general state spaces and show that it is in fact also necessary. As an application we obtain an explicit description of the limit in ergodic theorems for a suitable class of random transformations.
Paper Structure (5 sections, 10 theorems, 74 equations)

This paper contains 5 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strict irreducibility of Markov kernels
  4. Ergodicity of step skew products
  5. Random ergodic theorems

Key Result

theorem 3.1

Proposition. A Markov kernel $\pi$ is irreducible with respect to a $\pi$-invariant proba-bility measure $m$ if and only if the Markov operator $P$ corresponding to $\pi$ and $m$ is irreducible.

Theorems & Definitions (10)

  • theorem 3.1
  • theorem 3.2
  • theorem 3.3
  • theorem 3.4
  • theorem 3.5
  • theorem 4.1
  • theorem 4.2
  • theorem 4.3
  • theorem 5.1
  • theorem 5.2