%0 Generic %1 lopes2018maximum %A Lopes, Miles E. %D 2018 %K maximum multivariate normal range-test statistics %T On the Maximum of Dependent Gaussian Random Variables: A Sharp Bound for the Lower Tail %U http://arxiv.org/abs/1809.08539 %X Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. In the context of a finite index set, this paper offers such a bound, while also allowing for many types of dependence. Specifically, let $(X_1,\dots,X_n)$ be a centered Gaussian vector, with standardized entries, whose correlation matrix $R$ satisfies $\max_ij R_ij\rho_0$ for some constant $\rho_0(0,1)$. Then, for any $\epsilon_0ın (0,1-\rho_0)$, we establish an upper bound on the probability $P(\max_1in X_i\epsilon_02łog(n))$ that is a function of $\rho_0, \, \epsilon_0,$ and $n$. Furthermore, we show the bound is sharp, in the sense that it is attained up to a constant, for each $\rho_0$ and $\epsilon_0$.

@misc{lopes2018maximum, abstract = {Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. In the context of a finite index set, this paper offers such a bound, while also allowing for many types of dependence. Specifically, let $(X_1,\dots,X_n)$ be a centered Gaussian vector, with standardized entries, whose correlation matrix $R$ satisfies $\max_{i\neq j} R_{ij}\leq \rho_0$ for some constant $\rho_0\in (0,1)$. Then, for any $\epsilon_0\in (0,\sqrt{1-\rho_0})$, we establish an upper bound on the probability $\mathbb{P}(\max_{1\leq i\leq n} X_i\leq \epsilon_0\sqrt{2\log(n)})$ that is a function of $\rho_0, \, \epsilon_0,$ and $n$. Furthermore, we show the bound is sharp, in the sense that it is attained up to a constant, for each $\rho_0$ and $\epsilon_0$.}, added-at = {2019-10-16T07:27:08.000+0200}, author = {Lopes, Miles E.}, biburl = {https://www.bibsonomy.org/bibtex/27773bedc0727022190c032ce4138ad56/shabbychef}, description = {[1809.08539] On the Maximum of Dependent Gaussian Random Variables: A Sharp Bound for the Lower Tail}, interhash = {1f67b9cfbdac1b809ece9b1ae2b8116f}, intrahash = {7773bedc0727022190c032ce4138ad56}, keywords = {maximum multivariate normal range-test statistics}, note = {cite arxiv:1809.08539}, timestamp = {2019-10-16T07:27:08.000+0200}, title = {On the Maximum of Dependent Gaussian Random Variables: A Sharp Bound for the Lower Tail}, url = {http://arxiv.org/abs/1809.08539}, year = 2018 }