Sample Path Properties at Upcrossings

Abstract

Our main concern in the previous chapter has been the numbers and locations of upcrossings of high levels, and the relations between the upcrossings of several adjacent levels.For instance, we know from Theorem 9.3.2 and relation (9.2.3) that for a standard normal process each upcrossing of the high level u = u_τ; with a probability p = τ*/τ is accompanied by an upcrossing also of the level

$$u_{\tau * } = u - \frac{{\log p}} {u},$$

asymptotically independently of all other upcrossings of u _τ, and u_τ*.