On the compound Poisson limit for high level exceedances

Let ∊ 1 ∊ 2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un (τ)) of [3] for every τ > 0, where nP(∊ 1 > un (τ))→ τ . Assume only that P (there are j exceedances of un (τ) by ∊ 1, ∊ 2, · ··, ∊ n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.