An intermediate distribution between Gaussian and Cauchy distributions

TL;DR

This paper introduces a new intermediate distribution bridging Gaussian and Cauchy distributions, providing its density and characteristic functions, and explores applications in laser spectral line broadening and stock market returns.

Contribution

The paper constructs a novel intermediate distribution with explicit density and characteristic functions, and introduces weighted moments for distributions lacking moments.

Findings

Derived the probability density and characteristic functions of the intermediate distribution.

Applied the distribution to spectral line broadening in laser theory.

Utilized the distribution to model stock market returns.

Abstract

In this paper, we construct an intermediate distribution linking the Gaussian and the Cauchy distribution. We provide the probability density function and the corresponding characteristic function of the intermediate distribution. Because many kinds of distributions have no moment, we introduce weighted moments. Specifically, we consider weighted moments under two types of weighted functions: the cut-off function and the exponential function. Through these two types of weighted functions, we can obtain weighted moments for almost all distributions. We consider an application of the probability density function of the intermediate distribution on the spectral line broadening in laser theory. Moreover, we utilize the intermediate distribution to the problem of the stock market return in quantitative finance.