Partial summations of stationary sequences of non-Gaussian random variables

Abstract

The distribution of the sum of a finite number of identically distributed random variables is in many cases easily determined given that the variables are independent. The moments of any order of the sum can always be expressed by the moments of the single term without computational problems. However, in the case of dependency between the terms even calculation of a few of the first moments of the sum presents serious computational problems. By use of computerized symbol manipulations it is practicable to obtain exact moments of partial sums of stationary sequences of mutually dependent lognormal variables or polynomials of standard Gaussian variables. The dependency structure is induced by specifying the autocorrelation structure of the sequence of standard Gaussian variables. Particularly useful polynomials are the Winterstein approximations that distributionally fit with non-Gaussian variables up to the moments of the fourth order [Winterstein, S. R. Nonlinear vibration models for extremes and fatigue. J. Engng Mech. ASCE 114 (1988) 1772–1790] 1. A method to obtain the Winterstein approximation to a partial sum of a sequence of Winterstein approximations is explained and results are given for different autocorrelation functions of the generic Gaussian sequence. The primary purpose of the investigation is to provide a tool for judging the validity of the central limit theorem argument in specific applicational situations occurring in stochastic mechanics, that is, to judge the speed of convergence of the distribution of a sum (or an integral) of mutually dependent random variables to the Gaussian distribution. The paper is closely related to the work in Ditlevsen et al. [Ditlevsen, O., Mohr, G. & Hoffmeyer, P. Integration of non-Gaussian fields. Prob. Engng Mech 11 (1996) 15–23] 2.