Asymptotics On Powered-extremes And Normalized Maximum From Skew-normal Sequences

Convergence rate and high-order asymptotic expansion of distribution of order statis?tics are important parts in extreme value theory.This thesis is devoted to the investi-gation of asymptotics on powered-extremes and normalized maximum from skew-normal sequences.Let {M_n;n?1} denote the partial maximum of a sequence of independent random variables with common skew-normal distribution SN(?)with parameter A and|M_n|t be the powered-extremes with power index t>0.With the normalizing constants,we derive the rate of uniform convergence of skew-normal extremes and the higher-order expansion of the distribution of powered skew-normal extremes.It is shown that the value of power index t will affect the convergence rate of the distribution of |M_n|t.In the first part,we focus on the asymptotic behaviors of powered skew-normal ex-tremes.First,we determine the normalizing constants according to the sign of A and show that the limit distribution of normalized powered extreme of skew-normal distribu-tion is the Gumbel extreme value distribution.Then we establish higher-order expansions of the distribution of the powered extreme and investigate the optimal rate of convergence in this limit law.It is shown that with optimal normalizing constants the convergence rate of |M_n|t to its ultimate extreme value distribution is the order of 1/(logn)2 as t=2,and the convergence rate is the order of 1/logn for the case of 0<t?2.In the second part,the rate of uniform convergence of skew-normal extremes is derived.By choosing the normalizing constants according to the sign of A,the limit distribution of normalized maxima for the skew-normal distribution is derived.Then we investigate the rate of uniform convergence of skew-normal extremes.It is shown that with optimal normalizing constants the convergence rate of(M_n-bn)/an to its ultimate extreme value distribution is proportional to 1/logn.