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

Abstract

Let {Xi}i=1∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX1Xn+1, Sn = Σi=1nXi, and \(\bar X_n = \tfrac{{S_n }} {n} \). And let Nn be the point process formed by the exceedances of random level \((\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n \) 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].