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

TL;DR

This paper refines the understanding of the distribution of maxima in bivariate elliptical arrays by deriving second-order asymptotic expansions, considering the convergence rate of correlation and tail behavior.

Contribution

It provides second-order distributional expansions for maxima of elliptical arrays, improving upon previous first-order results by incorporating convergence rates and tail regular variation.

Findings

Derived second-order asymptotic expansions for maxima distributions.

Established the impact of convergence rate of correlation on maxima.

Enhanced previous results by including second-order regular variation effects.

Abstract

Let $\{ (\xi_{ni}, \eta_{ni}), 1\leq i \leq n, n\geq 1 \}$ be a triangular array of independent bivariate elliptical random vectors with the same distribution function as $(S_{1}, \rho_{n}S_{1}+\sqrt{1-\rho_{n}^2}S_{2})$, $\rho_{n}\in (0,1)$, 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 $\rho_{n}$ to $1$ is given. In this paper, under the refinement of the rate of convergence of $\rho_{n}$ 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.