Second-order Asymptotics on Distributions of Maxima of Bivariate Elliptical Arrays

Abstract

Let {(ξ ni , η ni ), 1 ≤ i ≤ n, n ≥ 1} be a triangular array of independent bivariate elliptical random vectors with the same distribution function as (S 1 , ρ n S 1 + 1 − ρ 2 n S 2 ), ρ n ∈ (0, 1), where (S 1 , S 2 ) is a bivariate spherical random vector. For the distribution function of radius S 2 1 + S 2 2 belonging to the max-domain of attraction of the Weibull distribution, Hashorva (2006) derived the limiting distribution of maximum of this triangular array if convergence rate of ρ n to 1 is given. In this paper, under the refinement of the rate of convergence of ρ n to 1 and the second-order regular variation of the distributional tail of radius, precise second-order distributional expansions of the normalized maxima of bivariate elliptical triangular arrays are established.