Second-order asymptotics on distributions of maxima of bivariate elliptical arrays

Let $\{ (ξ_{ni}, η_{ni}), 1\leq i \leq n, n\geq 1 \}$ be a triangular array of independent bivariate elliptical random vectors with the same distribution function as $(S_{1}, ρ_{n}S_{1}+\sqrt{1-ρ_{n}^2}S_{2})$, $ρ_{n}\in (0,1)$, where $(S_{1},S_{2})$ 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, 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.

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