The Characteristic Function of the Cube of a Gaussian Random Variable

TL;DR

This paper derives the characteristic function of the cube of a Gaussian random variable using spectral analysis of multiplication operators, providing a mathematical tool for understanding higher-order moments of Gaussian distributions.

Contribution

It introduces a novel spectral method to compute the characteristic function of the cube of a Gaussian, expanding analytical techniques in probability theory.

Findings

Explicit formula for the characteristic function of the cube of a Gaussian

Spectral resolution approach applied to probability distributions

Enhanced understanding of higher-order Gaussian moments

Abstract

Using the spectral resolution of the multiplication operator on the Schwartz class of $L^2(\mathbb{R},\mathbb{C})$, we compute the characteristic function of the cube of a Gaussian random variable.