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On the central limit question for strictly stationary, reversible Markov chains

Richard C. Bradley

Abstract

This paper will provide several classes of strictly stationary, countable-state, irreducible, aperiodic Markov chains that are reversible and have finite second moments, such that the central limit theorem fails to hold. The main purpose is to examine the extent to which, for the development of central limit theory for strictly stationary Markov chains (and functions of them) under the strong mixing and absolute regularity conditions, the property of reversibility (if it holds) can provide extra leverage. It is known, partly as a by-product of research done by Roberts, Rosenthal, and Tweedie in two papers in 1997 and 2001, that for the case of exponential mixing rates, reversibility provides notable extra leverage of that kind. In contrast, a class of counterexamples in a paper of Doukhan, Massart, and Rio in 1994 showed (implicitly) that for the case of power-type mixing rates, reversibility apparently provides almost no such extra leverage. Further perspective on that latter fact will be provided by some counterexamples in this paper. Other counterexamples here will (indirectly) provide some tentative, uncertain evidence for the possibility that for mixing rates that are ``between'' power-type and exponential (for example, sub-exponential), reversibility may in fact provide some small but nontrivial extra leverage.

On the central limit question for strictly stationary, reversible Markov chains

Abstract

This paper will provide several classes of strictly stationary, countable-state, irreducible, aperiodic Markov chains that are reversible and have finite second moments, such that the central limit theorem fails to hold. The main purpose is to examine the extent to which, for the development of central limit theory for strictly stationary Markov chains (and functions of them) under the strong mixing and absolute regularity conditions, the property of reversibility (if it holds) can provide extra leverage. It is known, partly as a by-product of research done by Roberts, Rosenthal, and Tweedie in two papers in 1997 and 2001, that for the case of exponential mixing rates, reversibility provides notable extra leverage of that kind. In contrast, a class of counterexamples in a paper of Doukhan, Massart, and Rio in 1994 showed (implicitly) that for the case of power-type mixing rates, reversibility apparently provides almost no such extra leverage. Further perspective on that latter fact will be provided by some counterexamples in this paper. Other counterexamples here will (indirectly) provide some tentative, uncertain evidence for the possibility that for mixing rates that are ``between'' power-type and exponential (for example, sub-exponential), reversibility may in fact provide some small but nontrivial extra leverage.

Paper Structure

This paper contains 10 sections, 20 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Mixing conditions and Markov chains
  3. The bounded case --- fast growth of variances
  4. The bounded case again --- very sharp mixing rates
  5. The unbounded case --- with quantiles
  6. Preliminaries
  7. Proof of Theorem \ref{['thm3.4']}
  8. Proof of Theorem \ref{['thm4.4']}
  9. Proof of Theorem \ref{['thm5.5']}
  10. Appendix

Key Result

Theorem 2.7

(Cuny and Lin). Suppose $X := (X_k, k \in \mathbb Z)$ is a real strictly stationary, reversible Markov chain that satisfies $\alpha$-mixing with $\alpha_X(n) \to 0$ at least exponentially fast as $n \to \infty$. Suppose $h: \mathbb R \to \mathbb R$ is a Borel function such that Then $\sigma^2 := E[\, [h(X_0)]^2] + 2 \cdot \sum_{k=1}^\infty E[h(X_0) \cdot h(X_k)]$ exists in $(0, \infty)$, with th

Theorems & Definitions (35)

  • Remark 2.4
  • Remark 2.5
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 3.2
  • Theorem 3.4
  • Remark 3.5
  • Theorem 4.1
  • Theorem 4.2
  • ...and 25 more