Optimal approximation by one Gaussian function to probability density functions

Abstract

In this paper, we introduce the optimal approximation by a Gaussian function for a probability density function. We show that the ap- proximation can be obtained by solving a non-linear system of parameters of Gaussian function. Then, to understand the non-normality of the em- pirical distributions observed in financial markets, we consider the nearly Gaussian function that consists of an optimally approximated Gaussian function and a small periodically oscillating density function. We show that, depending on the parameters of the oscillation, the nearly Gaussian functions can have fairly thick heavy tails.