The Rayleigh hypothesis for wave amplitudes and the LévyCramér
characterization of the Gaussian distribution
K. Bertram Broberg and Georg Lindgren
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2000
ISSN 14039338

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. For Gaussian
processes, the general form of its distribution is unknown, but the Rayleigh
distribution is often used as approximation; in fact, as the spectral width
of the process tends to zero, it becomes asymptotically Rayleigh.

For processes with general spectra, the amplitude distribution can be calculated
numerically with high accuracy. The aim of the present note is to illustrate
how the statistical dependence between the amplitude and the cycle
mean level affects the amplitude distribution. In particular a formal proof
is given for the fact that if, hypothetically, the cycle mean and the cycle
amplitude are stochastically independent and the expected number of upcrossings
follows Rice's formula, then the amplitude must be Rayleigh. The proof is
based on the LévyCramér characterization of the normal
distribution. However, as is known, the independence assumption is wrong.
The true dependence structure is illustrated for four different Gaussian
processes. For a lowfrequency white noise process, cycle mean is virtually
independent of amplitude, while for realistic ocean spectra there is a
considerable dependence.


