Irreducible Markov Chains

Abstract

In this chapter, we are interested in the mixing properties of irreducible Markov chains with continuous state space. More precisely, our aim is to give conditions implying strong mixing in the sense of Rosenblatt (1956) or $\beta$-mixing. Here we mainly focus on Markov chains which fail to be $\rho$-mixing (we refer to Bradley (1986) for a precise definition of $\rho$-mixing). Let us mention that $\rho$-mixing essentially needs a spectral gap condition in $L^2$ for the transition probability kernel. This condition is often too restrictive for the applications in econometric theory or nonparametric statistics.

Exercises

(1) Let $p>2$. Prove that, for any $a>1$ and any continuous distribution function F such that $\int _{\mathrm {I\!R}}|x|^p dF (x) < \infty$ and $\int _{\mathrm {I\!R}}x dF(x) = 0$, there exists a stationary sequence $\mathbb {Z}$ of random variables with common law F and $\beta$-mixing coefficients $\beta _i$ of the order of $i^{-a}$ such that

$$ {\mathrm {I\!E}}( |S_n|^p ) \ge c n^p \int _0^{n^{-a}} Q_0^p (u) du , $$

for some positive constant c. Compare this result with Theorem 6.3 and Corollary 6.1.

(2) Let $(\xi _i)_{i\in {\mathrm {Z\!Z}}}$ be a stationary Markov chain. Assume that the uniform mixing coefficients $\varphi _n$ converge to 0 as n tends to $\infty$. Prove that $\varphi _n = O (\rho ^n)$ for some $\rho$ in [0, 1[.