Tailoring the Gaussian Law for Excess Kurtosis and Skewness by Hermite Polynomials

This article aims at reshaping the normal law to account for tail-thickness and asymmetry, of which there is plenty of evidence in financial data. The inspiration to address the issue was provided by the orthogonality of Hermite polynomials with the Gaussian density as a weight function, with the Gram–Charlier expansion as background. A solution is then devised accordingly, by embodying skewness and excess-kurtosis in a normal kernel, via third- and forth-degree polynomial tune-up. Features of the densities so obtained are established in the main theorem of this article. In addition, a glance is cast at the issue of embodying between-squares correlation, and a solution is outlined.