Asymptotics and Bounds for Multivariate Gaussian Tails

Abstract

Let {X n , n 1} be a sequence of centered Gaussian random vectors in R d , d 2. In this paper we obtain asymptotic expansions (n → ∞) of the tail probability P{X n > t n } with t n ∈ R d a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.

Asymptotic properties of type I elliptical random vectors

“…In the light of Theorem 3.4 in Hashorva (2005a) there exists a unique non-empty index set M ⊂ K which defines the unique solution of P( −1 KK , a K ) such that…”

“…all the entries of the main diagonal of are equal 1. Lemma 2.1 implies then X i d = S 1 , 1 ≤ i ≤ k. As in the Gaussian case (see Hashorva 2005a) for the tail asymptotic expansion of interest the solution of the quadratic programming problem P( −1 , t n ) : minimise x 2 = x −1 x under the linear constraint x ≥ t n , (2.5) with t n a threshold inIR k is crucial. If the Savage condition (see Hashorva 2005a for more details)…”

“…1.1 with X a standard Gaussian random vectors is huge. We mention just few recent contributions Dai and Mukherjea (2001), Hüsler (2002a,b, 2003), Hashorva (2003Hashorva ( , 2005a.…”

Boundary Non-crossings of Brownian Pillow

“…Hashorva (2005b) shows the asymptotic behaviour (considering the Brownian bridge) of the corresponding discrete boundary non-crossing probability. we expect that our novel asymptotic result will have some implications for statistical applications.…”

Efficient Simulation for Dependent Rare Events with Applications to Extremes

“…The asymptotic properties of elliptical distributions also relate to this quadratic programming problem, which Hashorva [18,19] denotes as P(Σ −1 , t) := minimise x Σ −1 x under the linear constraint x ≥ t.…”