Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem
Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem
复制标题
中心极限定理某些条件的鞅近似和最优性
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发表时间：
2009
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通讯作者：
D. Volný
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作者：
D. Volný
AbstractLet (Xi) be a stationary and ergodic Markov chain with kernel Q and f an L2 function on its state space. If Q is a normal operator and f=(I−Q)1/2g (which is equivalent to the convergence of
$sum_{n=1}^{infty}frac{sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}$
in L2), 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)1/2L2, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to
$sum_{n=1}^{infty}c_{n}frac{|sum_{k=0}^{n-1}Q^{k}f|_{2}}{n^{3/2}}<infty$
for some sequence cn↘0, or by
$|sum_{k=0}^{n-1}Q^{k}f|_{2}=O(sqrt{n}/log n)$
, the CLT need not hold.
AbstractLet (Xi) be a stationary and ergodic Markov chain with kernel Q and f an L2 function on its state space. If Q is a normal operator and f=(I−Q)1/2g (which is equivalent to the convergence of
$sum_{n=1}^{infty}frac{sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}$
in L2), 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)1/2L2, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to
$sum_{n=1}^{infty}c_{n}frac{|sum_{k=0}^{n-1}Q^{k}f|_{2}}{n^{3/2}}<infty$
for some sequence cn↘0, or by
$|sum_{k=0}^{n-1}Q^{k}f|_{2}=O(sqrt{n}/log n)$
, the CLT need not hold.