On multivariate Gaussian tails

Abstract

Multivariate Mills ratio, Gaussian random sequences, tail asymptotics, quadratic programming,

A note on multivariate Gaussian estimates

“…In Section 3, we briefly review known approaches of multivariate Mills' ratio and give our main result of this paper. An example in Hashorva and Hüsler [3] is analyzed and show that for certain regions of parameters, our upper bound is better than those listed in their paper.…”

“…Next, we consider a simple example which was examined in detail in [3]. We show that the upper bound in (3.2) is better than those listed in [3] for certain range of parameters.…”

“…In Steck [9], two lower bounds were given by Jensen's inequality as outline in the introduction. In Hashorva and Hüsler [3], a pair of upper and lower bounds for multivariate Gaussian tails probability was given by the minimal and maximal eigenvalues after a shift substitution. They also obtained another lower bound by Jensen's inequality again after a shift substitution.…”

“…There is also a long history of studying them in various settings, in particular the case of multivariate Mills' ratio given in (3.1). Two well-known approaches are employed in the literature, see for example Hashorva and Hüsler [3]. One is translating multivariate Mills' ratio into a product of univariate Mills' ratios and then using corresponding lower and upper bounds of univariate Mills' ratio to estimate each term.…”

Tail Distribution of the Maximum of Correlated Gaussian Random Variables

“…Unfortunately, this is not the case. Assume, for example, that all pairs (X i , X j ) are jointly normal with zero mean, Var(X i ) = Var(X j ) = 1, and Cor(X i , X j ) = ρ i,j < 1, then using the results in Hashorva and Hüsler (2003) we can easily show that, for ease writing ρ = ρ i,j ,…”