Karhunen Loève expansion and distribution of non-Gaussian process maximum

In this note we show that when a second order random process is modeled through its truncated Karhunen Loève expansion and when the distribution of the random variables appearing in the expansion is approached by a Gaussian kernel, explicit relations for the mean number of up crossings, of the mean number of local maximums and more generally of Rice's moments can be derived in terms of Gaussian integrals. Several illustrations are given related to academic examples and natural hazards models. •Numerical expressions are given to compute the distribution of a process and its derivative.•They are based on Karhunen Loève expansion and Gaussian mixture model.•Application to Rice's formula is given.•Application to the calculation of extreme is given.•Several examples illustrate the effectiveness of the approach.