Page 485 - Invited Paper Session (IPS) - Volume 2

IPS355 Georg Lindgren

Thus, Rice’s formula for the average number of zero (up- or down-)
12
crossings is from 1939.
In “Random noise” Rice arrives at (3) by probabilistic arguments with a
stationary processes in mind. He also mentions that “equivalent results [on
the expected number of zeros] have been obtained independently by M.
Kac”. However, the context of the Kac formula is strictly the number of
5
roots to an algebraic equation, chosen at random under a variety of explicit
assumptions about its distribution. Due to the technical similarity of the
two formulas, the term “Kac-Rice” formula” is sometimes used.
b) Strict proofs, alternatives, and higher moments were late to be developed,
and not until the early1960s definite conditions for the validity were
2
published. The book by Cramer and Leadbetter from´ 1967 gives an
accurate account of the knowledge in the mid 1960s.
+
Consider first a stationary Gaussian process with mean zero. Then =

1 √ / − /2 0 is easily obtained from (2), with = ⋁(()), =
2
2 2 0 0 2
⋁( ()). When the process is non-differentiable = ∞. Ylvisaker 19
′
2
proved 1965 that the formula for holds, both for finite and for infinite
+

.
2
For the variance it took even longer and not until 1972 the final detail
4
was filled in by Don Geman. For the variance of the number of zeros of a
Gaussian process to be finite it is necessary (Geman) and sufficient (Cramer

and Leadbetter ) that ∫ ( () − (0))/ < ∞ for some δ > 0. In fact,
2
′′
′′
0
the result holds for any level.
c) “The problem of determining the distribution function for the distance
between two successive zerosseems to be quite difficult and apparently
nobody has as yet given a satisfactory solution”. This quote from Section
3.4 of “Random noise” still carries a grain of truth. The length of excursion
interval was the central theme of research following “Random noise”
during the first 15 years and it has gained renewed popularity from around
1990 from applications in physics and material science.
For the distribution of the length of an excursion above zero, Rice
suggested a Bonferroni type inclusion-exclusion series. The excursion time
density () approximated with the first three terms in the Rice in- and

exclusion series is

1
() ≈ { +− (0, ) − ∫ +− (0, , ) + ∬ +−−− (0, 1 , 2 , ) 1 2 }. (4)
+ 0 0 0
0 =0
0< 1 < 2 <

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