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

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 \(({S_1},{\rho _n}{S_1} + \sqrt {1 - \rho _n^2} {S_2}),{\rho _n} \in (0,1)\), where (S¹, S₂) is a bivariate spherical random vector. For the distribution function of radius \(\sqrt {S_1^2 + S_2^2} \) belonging to the max-domain of attraction of the Weibull distribution, the limiting distribution of maximum of this triangular array is known as the 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.