On multivariate Gaussian tails

Let {X _{n, n} ≥1} be a sequence of standard Gaussian random vectors in ℝ _d ,d ≥ 2. In this paper we derive lower and upper bounds for the tail probabilityP{X _n >t _n } witht _n ∈ ℝ _d some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962,J. Res. Nat. Bur. Standards Sect. B,66, 93–96). Furthermore, we prove exact asymptotics under fairly general conditions on bothX _n andt _n , as ‖t _n ‖→∞ where the correlation matrix Σ _n ofX _n may also depend onn.