Ratios and Cauchy Distribution

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.