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

K. Bertram Broberg and Georg Lindgren

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
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 Lévy-Cramér 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 low-frequency white noise process, cycle mean is virtually independent of amplitude, while for realistic ocean spectra there is a considerable dependence.