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arXiv 1005.4124math.PR
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方差非线性增长可逆过程的一个中心极限定理

A central limit theorem for reversible processes with non-linear growth of variance

Ou Zhao, Michael Woodroofe, Dalibor Volny

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11 pages
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摘要：

Kipnis和Varadhan表明，添加剂功能，$S_n$说，一个可逆的马尔可夫链的条件$E(S_n^{2})/n \to κ\in (0,\infty)$意味着收敛的条件分布的$S_n/\sqrt{E(S_n^{2}})$，给定的起点，以标准正态分布。我们在较弱的条件下重新讨论这个问题，$E(S_n^{2}) = n\ell(n)$，其中$\ell$是一个缓慢变化的函数。通过实例表明，在这种情况下，$S_n/\sqrt{E(S_n^{2}})$的条件分布不必收敛到标准正态分布，并给出了收敛到（可能是非标准的）正态分布的充分条件.

英文摘要：

Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a reversible Markov chain the condition $E(S_n^{2})/n \to κ\in (0,\infty)$ implies the convergence of the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, $E(S_n^{2}) = n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly non-standard) normal distribution are developed.

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