Comparison and anti-concentration bounds for maxima of Gaussian random vectors
Comparison and anti-concentration bounds for maxima of Gaussian random vectors
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DOI：
10.1007/s00440-014-0565-9
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发表时间：
2015-06-01
影响因子：
2
通讯作者：
Kato, Kengo
中科院分区：
文献类型：
--
作者：
Chernozhukov, Victor;Chetverikov, Denis;Kato, Kengo
Slepian型不等式和Sudakov-Fernique型不等式在概率论中，特别是在经验过程和极值理论中起着重要的作用，它们比较了高斯随机向量在协方差矩阵的某些限制下的最大期望.在这里，我们给出了明确的比较期望的光滑函数和分布函数的最大值的高斯随机向量的协方差矩阵没有任何限制。我们还建立了一个反浓度不等式的最大值的高斯随机向量，导出了一个有用的上限的L，vy浓度函数的高斯最大值。该界限是无量纲的，适用于具有任意协方差矩阵的向量。这个反集中不等式在建立高斯随机向量最大值之间的柯尔莫哥洛夫距离的界限中起着至关重要的作用。这些结果在数理统计中有直接的应用。作为应用的例子，我们建立了独立随机向量和的最大值的条件乘子中心极限定理，其中向量的维数可能远大于样本容量。
Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the L,vy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.