Upcrossing Probabilities for Stationary Gaussian Processes

Boundary Crossing Probabilities for Stationary Gaussian Processes and Brownian Motion

“…In that case our Ha(0) agrees with the 77a of Quails and Watanabe [10] and our result (3.4) is their Theorem 2.1. This result was obtained earlier by Pickands [12] under further restrictions on the incremental variance a2(t). Note that Pickands' definition of Xa(t) is slightly different from that of Qualls-Watanabe and his value for 77a also differs.…”

“…Our results take the form of a limit theorem: as a sequence of functions fn tend to infinity in an appropriate way we find constants A" such that A~xP(X(t) > f"(t), some t £\ T) -> 1. When T is a finite interval and the boundaries are without cusps, our results are a synthesis and extension of the work of Pickands [12], and Quails and Watanabe [10] on the one hand who assumed the^, to be constant, and Berman [2], [3] who considered translations of a fixed barrier, i.e. fn(t) = n + fit) on [0, T] with /(/) increasing.…”

“…Thus we need only consider the first term of (4.37). Pickands [12] has shown that H£(9) -^o^oo 77" ->oin Ha-Thus given e > 0, we may choose a and 9* such that |77aa(0) -77J < e for 9 > 9*. Thus the first term of (4.37) is greater than Now choose 9* so that the last term in (4.40) and (4.41) is small and then use the assumptions to see that the remaining terms are small for n > n0(9*).…”

“…For a ^ 1 or 2 it is interesting to note that the single constant 77a applies to all sets of boundaries with g = 0 (which always occurs when a < 1). We shall also use the following notation, which differs slightly from [12], [14]:…”

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“…di erentiable in the To avoid such problems special upcrossings, namely "-uprossings, are considered. We use the de nition given by Pickands (1969) for continuous processes. We also refer to Leadbetter, Lindgren and Rootz en (1983), Chapter 12, for more mathematical background.…”