Page 486 - Invited Paper Session (IPS) - Volume 2

IPS355 Georg Lindgren
Here +− (0, ) is the bivariate intensity that there is a zero upcrossing
0
at 0 and a downcrossing at t with no restriction to what happens in
between. That term obviously gives an overestimation of the density.
+−−
Subtracting the integral ∫ =0 0 (0, , ) that there is an extra
downcrossing somewhere ( ) gives an underestimation, and so on.
Improvements to the moment based Rice series was a recurrent theme in
10
applied studies of excursion times, during several decades.
d) Rice’s handling of crossing and multiple crossing intensities was quite
intuitive and the strict in-terpretation of “the distribution” of an excursion
was by no means clear. For example, one can ask for the conditional
probability that () > 0,0 < < , given that (0) = 0, upcrossing, an
event that has probability zero for any stationary Gaussian process. That
would give the probability that the excursion lasts more t time units after
an excursion-start at 0.
Kac-Slepian’s h.w. (horizontal window) conditioning (1959) gave a
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precise meaning to the conditioning on the zero-probability crossing
event. All conditional probabilities should be defined as the limit of a well
defined conditional probability, given a crossing, not exactly at 0, but in a
horizontal window [0, ℎ] the limit taken as h ↓ 0. Kac and Slepian also
showed, by an ergodic argument, that a h.w. conditioned distribution is
equal to the limit of the corresponding empirical distribution observed
after all −upcrossings > 0, tk as the observation interval goes to

infinitity: the probability that an excursion exceeds is

(( + ) > ; 0 < < |( ) = , h.w upcrossing)
0
0

#{ < ; ( + ) > , 0 < < } (#{ < 1; ( + ) > , 0 < < }
= lim = . (5)
→∞ #{ < } (#{ < 1})

Thus, the meaning of the condition is clear; what remains is to compute
the expectation in the nominator in (5). Rice backed away from the difficult
integrals involved in the higher order approximations in the Rice series (4).
Interestingly enough, he mentions the possibility to use the condition that
( + ) > for equally spaced points between 0 and t. He concludes

that also these integrals should be hard to evaluate. Indeed, advances in
statistical computing has made it possible to compute the expectation in
(5) with very high accuracy in reasonable time on a standard computer; see
Lindgren. 7
e) The 1967 book by Cramer and Leadbetter on stationary processes made
2
crossing problems, Rice’s formula, and its consequences available for the
general statistical community. The book also represented a link between
the moment based crossing analysis by Rice and others on the stream of

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