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首页> 外文会议>International Congress of Mathematicians Vol.1: Plenary Lectures and Ceremonies Aug 20-28, 2002 Beijing >Statistical Equivalence and Stochastic Process Limit Theorems
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Statistical Equivalence and Stochastic Process Limit Theorems

机译：统计等价和随机过程极限定理

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摘要

A classical limit theorem of stochastic process theory concerns the sample cumulative distribution function (CDF) from independent random variables. If the variables are uniformly distributed then these centered CDFs converge in a suitable sense to the sample paths of a Brownian Bridge. The so-called Hungarian construction of Koinlos, Major and Tusnady provides a strong form of this result. In this construction the CDFs and the Brownian Bridge sample paths are coupled through an appropriate representation of each on the same measurable space, and the convergence is uniform at a suitable rate. Within the last decade several asymptotic statistical-equivalence theorems for nonparametric problems have been proven, beginning with Brown and Low (1996) and Nussbaum (1996). The approach here to statistical-equivalence is firmly rooted within the asymptotic statistical theory created by L. Le Cam but in some respects goes beyond earlier results. This talk demonstrates the analogy between these results and those from the coupling method for proving stochastic process limit theorems. These two classes of theorems possess a strong inter-relationship, and technical methods from each domain can profitably be employed in the other. Results in a recent paper by Carter, Low. Zhang and myself will be described from this perspective.
机译：随机过程理论的经典极限定理涉及来自独立随机变量的样本累积分布函数（CDF）。如果变量是均匀分布的，则这些居中的CDF在合适的意义上收敛到布朗桥的采样路径。所谓的Koinlos，Major和Tusnady的匈牙利建筑提供了这种结果的有力形式。在这种结构中，CDF和布朗桥采样路径通过在相同的可测量空间上通过各自的适当表示耦合在一起，并且收敛以适当的速率均匀。在过去的十年中，从Brown和Low（1996）和Nussbaum（1996）开始，已经证明了几种非参数渐近统计等价定理。统计等效性的方法牢固地扎根于L. Le Cam创建的渐近统计理论，但在某些方面超出了先前的结果。演讲证明了这些结果与耦合方法的相似性，以证明随机过程极限定理。这两类定理具有很强的相互关系，并且每个领域的技术方法都可以在另一个领域中有利地采用。结果来自Carter，Low的最新论文。张和我将从这个角度来描述。

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