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高维概率

High Dimensional Probability

Evarist Giné, Vladimir Koltchinskii, Wenbo Li, Joel Zinn

备注：
Published at http://dx.doi.org/10.1214/074921706000000905 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
IMS Lecture Notes Monograph Series 2006, Vol. 51, i-vii
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摘要：

大约40年前，几位研究人员意识到，如果在不强加与手头问题无关的结构的情况下考虑某些概率论对象的基本特征，特别是高斯过程和极限定理，可能会更好地理解它们。例如，在高斯过程的样本连续和有界性的情况下，本质特征是由过程的协方差结构在索引集上诱导的度量或伪度量结构，而不管索引集可能是什么。这一观点最终导致了Fernique-Talagrand优化测度刻画样本有界性和高斯过程的连续性，从而解决了Kolmogorov提出的一个重要问题。同样，可分的巴拿赫空间为大数定律、中心极限定理和重对数定律提供了一个最小的设置，这导致了对空间的最小（必要和/或充分）几何性质的阐明，在这些几何性质下，这些定理的不同形式都成立。然而，鉴于对经验过程的重新兴趣，这一主题对现代统计学产生了相当大的影响，人们不得不处理不可分的巴拿赫空间，即$\mathcal{L}_{\infty}$。随着可分性的抛弃，为高斯过程和可分巴拿赫空间中的极限定理和不等式而发展的技术，连同组合技术，导致了一般指标集上的独立有界过程的和的强大的不等式和极限定理，或者换句话说，为一般经验过程。

英文摘要：

About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings that do not impose structures extraneous to the problems at hand. For instance, in the case of sample continuity and boundedness of Gaussian processes, the essential feature is the metric or pseudometric structure induced on the index set by the covariance structure of the process, regardless of what the index set may be. This point of view ultimately led to the Fernique-Talagrand majorizing measure characterization of sample boundedness and continuity of Gaussian processes, thus solving an important problem posed by Kolmogorov. Similarly, separable Banach spaces provided a minimal setting for the law of large numbers, the central limit theorem and the law of the iterated logarithm, and this led to the elucidation of the minimal (necessary and/or sufficient) geometric properties of the space under which different forms of these theorems hold. However, in light of renewed interest in Empirical processes, a subject that has considerably influenced modern Statistics, one had to deal with a non-separable Banach space, namely $\mathcal{L}_{\infty}$. With separability discarded, the techniques developed for Gaussian processes and for limit theorems and inequalities in separable Banach spaces, together with combinatorial techniques, led to powerful inequalities and limit theorems for sums of independent bounded processes over general index sets, or, in other words, for general empirical processes.

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