Remarks on compound Poisson approximation of Gaussian random sequences

Abstract

Let {X i, i⩾1} be a sequence of m-dependent stationary standard Gaussian random variables and u n, n∈ N some positive constants. In this note we generalise results of Raab (Extremes 1(3) (1999) 29.), who considered compound Poisson approximation for W n=∑ i=1 n 1 {X i>u n} the number of exceedances above the level u n . More precisely, the main result concerns an upper asymptotic bound for the total variational distance d TV (W n, CP(λ ∗)) where CP(λ ∗) = d N 1+2N 2+⋯+rN r, with 2⩽ r⩽2 m and N i = d Poi(λ i), λ i>0 are independent Poisson random variables.