On the calculation of the statistical distribution of maxima of slightly non-linear,quadratic,stationary stochastic variables

Abstract

A method for calculation of the statistical distribution of maxima of stochastic variables of the type: $$\begin{array}{c}\mathrm{Y}={\mathit{\alpha }}_i{X}_i+a_{ij}{X}_i{X}_j\end{array}$$ where {a_ij} is ‘small’ and {X_i} is a Gaussian distributed stationary stochastic process, is shown. This method is based on an asymptotic series expansion of the probability density function for $\{a_{ij}\}\to 0$ , which is compared to the exact solution for: $$\begin{array}{c}\mathrm{Y}=\mathrm{X}+\mathit{\epsilon }{X}^2\end{array}$$ and the validity of the asymptotic series is discussed.

The statistical distribution of the combined stresses at the deck corner of a ship is calculated according to the present method and some characteristic parameters of the extreme-value statistics are given. These parameters are compared to some standard estimates, which are found to be conservative. At last the possibility of calculating the long-term distribution of maxima of non-linear variables is discussed.