Integration of non-Gaussian fields

Abstract

The limitations of the validity of the central limit theorem argument as applied to definite integrals of non-Gaussian random fields are empirically explored by way of examples. The purpose is to investigate in specific cases whether the asymptotic convergence to the Gaussian distribution is fast enough to justify that it is sufficiently accurate for the applications to shortcut the problem and just assume that the distribution of the relevant stochastic integral is Gaussian. An earlier published example exhibiting this problem concerns silo pressure fields. [Ditlevsen, O., Christensen, C. and Randrup-Thomsen, S. Reliability of silo ring under lognormal stochastic pressure using stochastic interpolation. Proc. IUTAM Symp., Probabilistic Structural Mechanics: Advances in Structural Reliability Methods, San Antonio, TX, USA, June 1993 (eds.: P. D. Spanos & Y.-T. Wu) pp. 134–162. Springer, Berlin, 1994] 1 The numerical technique applied to obtain approximate information about the distribution of the integral is based on a recursive application of Winterstein approximations (moment fitted linear combinations of Hermite polynomials of standard Gaussian variables). The method uses the very long exact formulas for the 3rd and 4th moments of any linear combination of two correlated four-term Winterstein approximations. These formulas are derived by computerized symbol manipulations. Some of the results are compared with some special exact results for sums of Winterstein approximations. [Mohr, G. & Didevsen, O. Partial summations of stationary sequences of Winterstein approximations. Prob. Engng Mech. 11 (1996) 25–30.] 2 For decreasing correlation extension including negative correlation, problems of increasing sensitivity to the recursive approximations show up. For practical use of the method, it may therefore, in special situations with negative correlation, be necessary to introduce numerical integration checks or simulation checks of the results.