A note on the statistics of ordered peaks in stationary stochastic processes

Numerical simulation experiments have been carried out to evaluate the relative performance of three approximate formulations owing to Amini and Trifunac,[1] Gupta and Trifunac[2] and Basu et al.[3] for probability distributions of ordered peaks in stationary stochastic processes. The first two formulations are based on the assumption that the unordered peaks are statistically independent; whereas the formulation of Basu et al.[3] considers the dependence via Markov transition probability. In the formulation of Gupta and Trifunac,[2] the probability distribution of nth order peak is inherently conditioned by the fact that at most (n−1) peaks can occur with higher amplitudes and, thus, the ordered peaks are necessarily dependent. In this paper, ensembles of stationary time-histories are generated for three PSDFs representing a narrow-band, a broad-band and a band-limited white noise type of processes. Comparison of the results for the expected value and standard deviation for various orders of peaks, obtained by averaging over these ensembles, with the corresponding results obtained from the above mentioned three approximations, defined in terms of the moments of the PSDFs, has shown that the formulation of Gupta and Trifunac[2] describes the data well.