Evaluation of Densities and Distributions via Hermite and Generalized Laguerre Series Employing High-Order Expansion Coefficients Determined Recursively via Moments or Cumulants

: In the theoretical analysis of performance of some systems with nonlinearities and/or memory, it often happens that the only statistics about the decision ( or output) random variable of interest that can be easily found are the moments, or in other cases, the cumulants. Explicit relations for the low-order expansion coefficients in Edgeworth of Gram-Charlier series are available in terms of the available moments or cumulants. However, for higher- order moments and cumulants, these explicit nonrecursive relations are very tedious to derive, become extremely lengthy, and are not practical to use. This document addresses the problem of obtaining accurate high-order series expansion approximations of the probability density function and cumulative distribution function of a random variable of interest, in terms of the available moments or cumulants of that random available. The necessity of being available to approximate probability density functions and cumulative distribution functions from knowledge of either moments for the cumulants, is that some physical problems have these particular statistics as natural and convenient starting points.