Multivariate probability distribution of ordered peaks of vector Gaussian random processes

The problem of determining the joint probability distribution of ordered peaks of jointly stationary Gaussian random processes is considered. The solution is obtained by modeling the number of times a specified threshold is crossed by the component processes as a multivariate Poisson process. Based on this, the joint probability distribution of the time required for the nth crossing of a specified level with positive slope is derived. This formulation is further extended to derive the joint distribution of ordered peaks in a given time interval. An illustrative example on a bivariate Gaussian random process is presented and the analytical predictions are shown to compare reasonably well with corresponding results from Monte Carlo simulations. Also presented is an analysis of response of a randomly driven multi-degree of freedom system with emphasis on the sensitivity of ordered peak characteristics with respect to changes in system parameters. It is demonstrated that higher order statistics are generally more sensitive to changes in system characteristics—a property that has potential for application in structural model updating and damage detection.