Ratios and Cauchy Distribution

TL;DR

This paper explores the properties of ratios of Gaussian variables and their convex combinations, revealing new conditions under which these combinations follow a Cauchy distribution and providing methods to construct Cauchy copulas.

Contribution

It generalizes known results about ratios of Gaussian variables to broader settings, including dependent and non-normal cases, and introduces new ways to construct Cauchy copulas.

Findings

Convex combinations of ratios of Gaussian vectors are Cauchy distributed.

Independence and normality are not necessary for the ratios to be Cauchy.

The results suggest joint normality may be the only case where the variables are independent.

Abstract

It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent joint work, the author proved a surprising multivariate generalization of the above facts. Fix $m > 1$ and let $\Sigma$ be a $m\times m$ positive semi-definite matrix. Let $X,Y \sim \mathrm{N}(0,\Sigma)$ be independent vectors. Let $\vec{w}=(w_1, \dots, w_m)$ be a vector of non-negative numbers with $\sum_{j=1}^m w_j = 1.$ The author proved recently that the random variable $$ Z = \sum_{j=1}^m w_j\frac{X_j}{Y_j}\; $$ also has the standard Cauchy distribution. In this note, we provide some more understanding of this result and give a number of natural generalizations. In particular, we observe that if $(X,Y)$ have the same marginal distribution, they need neither be independent nor be jointly normal for $Z$ to be Cauchy distributed. In fact, our calculations suggest that joint normality of $(X,Y)$ may be the only instance in which they can be independent. Our results also give a method to construct copulas of Cauchy distributions.