The Asymptotic Properties Of Extremes From Mixed Chi-square Distribution

This paper mainly gives a detailed description on the asymptotic properties of extremes of mixed chi-square distribution under linear and power normalization respectively.Let Fx(x)be the cumulative distribution function of mixed chi-square distribution,we have Fx(x)=P1F1(x)+P2F2(x)+…+PrFr(x),of which Fm(x)represents the corresponding chi-square distribution function with 1 ≤ m ≤ r.In this paper,we derive the asymptotic distribution of maximum value of mixed chi-square distribution sequence and the pointwise convergence rate under linear normalization.Besides,the asymptotic expansion of F_x~n(·)is given under an optimal choice of norming constant.Similarly,the asymptotic expansion of F_x~n(·)is derived under power normalization.Part 1 focuses on the asymptotic properties of the distribution of normalized maximum from mixed chi-square distribution under linear normalization,details see chapter 3.First of all,this part gives two conclusions about the chi-square distribution:Mills’ type ratios and the tail representation.These conclusions can help us derive two facts that Fx∈D1(Λ)and two kinds of norming constants,so that the pointwise convergence rate of the distribution of its partial maximum to the Gumbel extreme value distribution is derived under linear normalization.Finally,with precise tail representation,the asymptotic expansion of the distribution of normalized maximum is given under an optimal choice of norming constants.Part 2 focuses on the asymptotic expansion of the distribution of normalized maximum from mixed chi-square distribution under power normalization,details see chapter 4.Firstly,the fact that Fx∈Dp(Φ1)under power normalization can be found by the domain of attraction of Fx under linear normalization,so we can obtain the optimal choice of norming constant.Then,the asymptotic expansion of the distribution of normalized maximum under power normalization will be derived.