Seminarium  i matematisk statistik fredagen den 13 oktober 13.15 i MH227


Georg Lindgren, Matematisk statistik, Lunds universitet


The Rayleigh hypothesis for wave amplitudes and the Lévy-Cramér
characterization of the Gaussian distribution

ABSTRACT

The distribution of cycle amplitude, i.e. the vertical distance between a local
maximum and the following local minimum, in a continuous stochastic process is
an important quantity in many engineering fields. If the process is Gaussian a
common belief is that the amplitude has a Rayleigh distribution; and in fact,
as the spectral width of the process tends to zero, it becomes asymptotically
Rayleigh.

It has been known since long that if the cycle mean and the cycle amplitude are
stochastically independent and the expected number of upcrossings follows
Rice's formula, then in fact the amplitude must be Rayleigh. It has also been
known, almost as long, that the independence assumption is wrong. The aim of
this note is to present a proof  given by Bertram Broberg (1961) of the
Rayleigh implication by means of the Lévy-Cramér characterization of the normal
distribution, and to illustrate the true dependence structure for three
different Gaussian processes. For a low-frequency white noise process cycle
mean is virtually independent of amplitude, while for realistic ocean spectra
there is a considerable dependence.

The problem was the origin of my interest in random waves in the sixties and
could be of some historical interest – it is also a nice example of the
Lévy-Cramér characterization: if the sum of two independent random variables
has a normal distribution, then the terms are themselves normal. (Olav
Kallenberg, previous lecturer in Lund, proved the counterpart for the Poisson
distribution.)