Page 487 - Invited Paper Session (IPS) - Volume 2

IPS355 Georg Lindgren
crossings, and the emerging extreme value theory for dependent
sequences and processes. One of its most important theorems says that
under conditions that restrict local clumping and dependence over long
distances the stream of level upcrossings form an asymptotic Poisson
process as the level increases. The proof given in Cramer and Leadbetter 2
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leans heavily on computations in Rice and points forward to modern
extremal mixing conditions.
Rice’s formula together with the Poisson approximation for high level
upcrossings is often used in reliability analysis to approximate the
probability of the occurrence of an extreme value in a differentiable
+
stationary load process: P (no upcrossing of level before time ) ≈ − .

5. Influence of “Random noise” on stochastic modelling
The comprehensive treatment of random noise in communication systems
in “Random noise” spurred stochastic research also in applied physics,
acoustics, and material science, but it had remarkably little effect on statistics
and probability. A rare exception is M.S. Bartlett’s book from 1955 on methods
1
and applications of stochastic processes, focusing on applications. Bartlett
mentions Rice, en passant, in connection with the recurrence time of level
crossings.
Much more “statistically significant” is the dramatic effect that “Random
noise” had on physical oceanography and naval engineering. In the early 1950s
statistical studies of ocean waves concentrated on wave height distributions,
time correlation, and Fourier analysis. In an unusually fortunate cooperation
between Willard Pierson, an oceanographer, and Manley St Denis, a naval
architect (ship builder), wrote a pioneering paper on the motion of ship on a
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stochastic ocean. Heavily inspired by Rice’s “Random noise” they developed
a full Gaussian model in time and 2D space for irregular waves, generalizing
(1) to elementary waves of different wavelengths coming from different
directions. They also implemented filtering techniques to describe the
movements of a ship sailing on the ocean and derived the necessary filter
functions.
Michael Longuet-Higgins was a British oceanographer who published more
than twenty articles on statistical properties of waves between 1952 and 1991.
9
A paper from 1957 is a parallel to Rice’s “Random noise” in its detailed
description of stochastic characteristics of a random sea surface. His analysis
1962 10 of Rice’s in- and exclusion series for the distribution of intervals
between zeros in a Gaussian process stretches the techniques as far as can be
expected; further exact results had to wait until numerical algorithms and
computer technology had advanced. 7

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