Crossings of Non‐Gaussian Translation Processes

Abstract

Mean upcrossing rates are determined for translation processes obtained from normal processes by univariate, nonlinear transformations. Monotonic and more general transformations are studied. It is shown that translation processes can have any marginal distribution and autocorrelation function and that approximations proposed previously for the mean upcrossing rate of non‐Gaussian processes can be unsatisfactory. These approximations assume that the process and its time‐derivative, considered to follow a Gaussian distribution, are independent. Theoretical findings are applied to determine crossing characteristics of wind speeds, river flows, and other non‐Guassian processes.