Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Dalibor Volný ()
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Dalibor Volný: Université de Rouen

Journal of Theoretical Probability, 2010, vol. 23, issue 3, 888-903

Abstract: Abstract Let (X i ) be a stationary and ergodic Markov chain with kernel Q and 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. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lifšic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of $\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}$ , in particular by $\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)$ , q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(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{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}

Keywords: Martingale approximation; Martingale difference sequence; Strictly stationary process; Markov chain; Central limit theorem; 60G10; 60G42; 28D05; 60F05 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10959-010-0275-x

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