Martingale approximation and optimality of some conditions for the central limit theorem

Abstract

Let $(X_i)$ be a stationary and ergodic Markov chain with kernel $Q$, $f$ an $L^2$ function on its state space. If $Q$ is a normal operator and $f=(I-Q{)}^{1/2}g$ (which is equivalent to the convergence of ${\sum }_{n=1}^{\infty }\frac{{\sum }_{k=0}^{n-1}{Q}^{k}f}{n^{3/2}}$ in $L^2$), we have the central limit theorem (cf\. \cite{D-L 1}, \cite{G-L 2}). Without assuming normality of $Q$, the CLT is implied by the convergence of ${\sum }_{n=1}^{\infty }\frac{\Vert {\sum }_{k=0}^{n-1}{Q}^{k}f{\Vert }_2}{n^{3/2}}$, in particular by $\Vert {\sum }_{k=0}^{n-1}{Q}^{k}f{\Vert }_2=o(\sqrt{n}/{\log }^{q}n)$, $q>1$ by \cite{M-Wu} and \cite{Wu-Wo} respectively. We shall show that if $Q$ is not normal and $f\in (I-Q{)}^{1/2}{L}^2$, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to ${\sum }_{n=1}^{\infty }{c}_n\frac{\Vert {\sum }_{k=0}^{n-1}{Q}^{k}f{\Vert }_2}{n^{3/2}}<\infty$ for some sequence $c_n\searrow 0$, or by $\Vert {\sum }_{k=0}^{n-1}{Q}^{k}f{\Vert }_2=O(\sqrt{n}/\log n)$, the CLT need not hold.

to appear in Journal of Theoretical Probability

References in corpus (1)

• Pointwise ergodic theorems with rate and application to limit theorems for stationary processes