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

Abstract

Let {X _i } ^∞ _i=1 be a standardized stationary Gaussian sequence with covariance function r(n) = EX ₁ X _n+1, S _n = Σ ⁿ _i=1 X _i , and $\bar X_n = \tfrac{{S_n }} {n}$. And let N _n 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 X ₁,X ₂,…, X _n . Under some mild conditions, N _n and S _n are asymptotically independent, and N _n converges weakly to a Poisson process on (0,1].