Asymptotic Properties Of Extreme From The Generalized Short-tailed Symmetric Distribution

In this paper,we study the asymptotic behavior of normalized maximum from a sequence of independent and identically distributed random variable with the generalized short-tailed symmetric distribution.This paper is divided into three parts:In the first part,firstly,we give the tail expression of the generalized short-tailed symmetric distribution.Secondly,the asymptotic expansions of distribution and density of normalized maximum from generalized short-tailed symmetric distribution are obtained.Thirdly,the associated convergence rates of the distribution and density of the extreme to its extreme value distribution are derived.In the second partusing the tail expression in the first part and the asymptotic expansions of distribution and density of maximum from generalized short-tailed symmetric distribution,the associated convergence of the distribution and density of the extreme to its extreme value distribution are established under the power normalization.In the third part,applying the asymptotic expansion of the distribution function in the first part,the asymptotic expansion of the moments of normalized maximum from generalized short-tailed symmetric distribution is obtained.