Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

摘要

Let {Xi}∞i=1 be a standardized stationary Gaussian sequence with covariance function r(n) =EX1X(n+1),Sn=Σni=1 Xi,and Xn =Sn/n.And let Nn be the point process formed by the exceedances of random level(x/(2logn)1/2+(2logn1/2)-log(4πlogn)/(2logn)1/2 [1-r(n)]1/2+Xn by X1,X2,...,Xn.Under some mild conditions,Nn and Sn are asymptotically independent,and Nn converges weakly to a Poisson process on(0,1].