A related concept are point processes of s-upcrossings.
Poisson convergence of the point process of s-upcrossings may be obtained (Leadbetter et al
For an a. s. continuous process Y an s-upcrossing of level u is a point t0 with Y (t) < u when t ∈ (t0 − s, t0) and Y (t0) = u.
Every s-upcrossing is an upcrossing, while obviously an upcrossing need not to be an s-upcrossing.
If the assumption of Theorem 1.4.5 holds, so that P (1) = 1 and P (2) = 0, the kernel function f is non-increasing, and the driving Lévy process is a positive compound Poisson process, then also the point process of upcrossings and s-upcrossings con- verges to a Poisson process since upcrossings and s-upcrossings occur only at positive jump times of the Lévy process in combination with the supremum of the kernel function (Theorem 1.4.5, Corollary 1.4.2).
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