高斯过程相应不等式是一个真正的高斯过程基本工具, 类似于切比雪夫的统计不等式和鞅理论中的极大不等式, 甚至因其简单性和强大功能在适用性上显得更为出色. 本文主要介绍两个高斯不等式, 即Borell–TIS不等式和比较不等式(Slepian不等式). 本文第一章阐述了高斯过程的定义和相关性质, 并通过讨论几乎处处有限和/或连续的充分条件引出相关定理, 为后续提供理论基础.第二章详细地介绍了Borell-TIS不等式及其相关结论, 然后引用三个引理并通过特殊到一般的方法完成Borell-TIS不等式的证明. 对于任意中心化的连续高斯场, Borell-TIS不等式为游弋概率提供了一个通用的界限. 第三章的主要内容则为Slepian不等式及其最重要的延伸——Sudakov-Fernique不等式. 比较不等式同样是非常重要而且基础的理论, 可以通过高斯场的协方差函数之间的关系来比较游弋概率和上确界的期望.

The Gaussian inequality is a truly basic tool of Gaussian processes, somewhat akin to Chebyshev’s inequality in statistics or maximal inequalities in martingale theory, and even better for its simplicity and powerful function in applicability. This article is mainly about two Gaussian inequalities, the Borell–TIS (Borell–Tsirelson–Ibragimov–Sudakov) inequality and comparison inequalities (Slepian’s inequality). Chapter 1 is mainly about the definition and some habitudes of the Gaussian processes, which also shows some theorems for later by discussing when are Gaussian processes almost surely bounded and/or continuous. And then, chapter 2 describes the Borell-TIS inequality and its relatives in detail, and then complete its proof by three lemmas and lifting result from finite to general. The Borell–TIS inequality gives a universal bound for the excursion probability for any centered, continuous Gaussian field. The main result of Chapter 3 is Slepian’s inequality and its most important extension, Sudakov-Fernique inequality. Comparison inequality and its relatives are just as important and basic, and allow one to use relationships between covariance functions of Gaussian fields to compare excursion probabilities and expectations of suprema.