Abstract
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.
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Hashorva, E., Hüsler, J. On multivariate Gaussian tails. Ann Inst Stat Math 55, 507–522 (2003). https://doi.org/10.1007/BF02517804
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DOI: https://doi.org/10.1007/BF02517804