Asymptotics On Powered-Extremes Of Gaussian Sequences

This thesis mainly investigates asymptotic behaviors of powered extremes of Gaussian samples.Let Xn,n denote the maximum of Gaussian sequence and |Xn,n|~t be the powered-extremes with power index t>0.We have the following find-ings:Distributional expansions and convergence rates of powered-extremes formed by univariate Gaussian sequence can be affected along with different power index t,contrary to the case of bivariate Gaussian triangular arrays.The latter shows that the higher-order expansions and convergence rates of powered-extremes has no relation to the power index t.The thesis is organized as the following two parts.In the first part,we mainly focus on the asymptotics of powered-extremes of univariate Gaussian sequence.Firstly,with power index t>0,higher-order expansions of distributions of powered-extremes are established under the optimal normalizing constants,which shows that the distributions of powered extremes con-verges to Gumbel distributions and convergence rate is proportional to 1/logn.However,with power index t = 2,the convergence rates can be improved to be of the order of 1/(logn)~2. Subsequently,based on the results mentioned above,asymptotic expansions of densities of powered-extremes are derived.For the second part,the second-order expansions of the joint distributions of powered maxima for a triangular array of bivariate Gaussian random vectors are considered.Under the Hiisler-Reiss condition,joint limit distribution of powered maxima is the Hiisler-Reiss max-stable distribution.Furthermore,the second-order expansions of the joint distributions of powered maxima are established under the refined Husler-Reiss condition.The main results show that the choices of power index t no longer improves the convergence rate of the joint distribution to its limit,which is still of the order of 1/log n.