arXiv 0708.41832007-08math.PR

关于鞅逼近

On martingale approximations

Ou Zhao, Michael Woodroofe

备注：: Published in at http://dx.doi.org/10.1214/07-AAP505 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Applied Probability 2008, Vol. 18, No. 5, 1831-1847
DOI：: 10.1214/07-AAP505

摘要：

考虑马尔可夫链$W_k$的可加泛函，其平稳（边缘）分布和转移函数由$π$和$Q$表示，例如$S_n=g(W_1)+...+g(W_n)$，其中$g$是平方可积的，并且关于$π$的均值为0。如果$S_n$有$S_n=M_n+R_n$的形式，其中$M_n$是一个具有平稳增量和$E(R_n^2)=o(n)$的平方可积鞅，则称$g$允许鞅逼近。这种近似的必要和充分条件的开发。两个明显的必要条件是$E[E(S_n|W_1)^2]=o(n)$和$\lim_{n\to \infty}E(S_n^2)/n<\infty$。假设其中的第一个，设$\Vert g\Vert^2_+=\limsup_{n\to \infty}E(S_n^2)/n$ ;则$\Vert\cdot\Vert_+$在$L^2(π)$的子空间上定义一个伪范数，其中它是有限的。在一个主要结果中，一个简单的必要条件和充分条件的鞅逼近的$\Vert\cdot\Vert_+$。令$Q^*$表示$Q$的伴随算子，将其视为从$L^2(π)$到自身的线性算子，并考虑余等距（$QQ^*=I$），这是包括移位过程的重要特殊情况。在另一个主要的结果是一个方便的正交规范的基础$L_0^2(π)$确定连同一个简单的必要和充分条件的鞅逼近的存在方面的系数的$g$的扩展相对于这个基础。

英文摘要：

Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $π$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $π$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n|W_1)^2]=o(n)$ and $\lim_{n\to \infty}E(S_n^2)/n<\infty$. Assuming the first of these, let $\Vert g\Vert^2_+=\limsup_{n\to \infty}E(S_n^2)/n$; then $\Vert\cdot\Vert_+$ defines a pseudo norm on the subspace of $L^2(π)$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\Vert\cdot\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(π)$ into itself, and consider co-isometries ($QQ^*=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(π)$ is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of $g$ with respect to this basis.

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摘要：

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