Generalization of the Edgeworth and Gram-Charlier series and quasi-probability densities (Wavelet analysis and signal processing)

Abstract

The main historical purpose of the standard Gram-Charlier and Edgeworth series is to " correct" a Gaussian distributions when new information, such as moments, is given which does not match those of the Gaussian. These methods relate a probability distribution to a Gaussian by way of an operator transformation that is a function of the differentiation operator. We use the methods of the phase-space formulation of quantum mechanics to generalize these methods. We generalize in two ways. First, we relate any two probability distributions by way differentiation operator. Second, we generalize to the case where the differentiation operator is replaced by an arbitrary Hermitian operator. The generalization results in a unified approach for the operator transformation of probability densities. Also, when the Edgeworth and Gram-Charlier series are truncated, the resulting approximation is generally not manifestly positive. We present methods where the truncated series remain manifestly positive.