Bounding the maximum of dependent random variables

Abstract

Let $M_{n}$ be the maximum of $n$ unit Gaussian variables $X_{1},\ldots,X_{n}$ with correlation matrix having minimum eigenvalue $\lambda_{n}$. Then \[M_{n}\ge\lambda_{n}\sqrt{2\log n}+o_{p}(1).\] As an application, the log likelihood ratio statistic testing for the presence of two components in a 1-dimensional exponential family mixture, with one component known, is shown to be at least $\frac{1} {2}\log\log n(1+o_{p}(n))$ under the null hypothesis that there is only one component.