二元椭圆阵极大值分布的二阶渐近性
Second-order asymptotics on distributions of maxima of bivariate elliptical arrays
摘要：
令$\{ (ξ_{ni}, η_{ni}), 1\leq i \leq n, n\geq 1 \}$是具有与$(S_{1}, ρ_{n}S_{1}+\sqrt{1-ρ_{n}^2}S_{2})$相同的分布函数的独立二元椭圆随机向量的三角阵列，$ρ_{n}\in (0,1)$，其中$(S_{1},S_{2})$是二元球形随机向量。对于半径为$\sqrt{S_{1}^2+S_{2}^2}$的分布函数属于韦布尔分布的最大吸引区域，如果给定$ρ_{n}$到$1$的收敛速度，Hashorva（2006）导出了该三角形阵列的最大值的极限分布。本文通过对$ρ_{n}$到$1$收敛速度的精化和半径分布尾的二阶正则变化，建立了二元椭圆三角阵归一化极大值的精确二阶分布展开式。
英文摘要：
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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