Joint behavior of point process of exceedances and partial sum from a Gaussian sequence

Consider a triangular array of mean zero Gaussian random variables. Under some weak conditions this paper proves that the partial sums and the point processes of exceedances formed by the array are asymptotically independent. For a standardized stationary Gaussian sequence, it is shown under some mild conditions that the point process of exceedances formed by the sequence (after centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. Finally, the joint limiting distributions of the extreme order statistics and the partial sums are obtained.