A Uniqueness Property Not Enjoyed by the Normal Distribution

Abstract

It is well known that if $X$ and $Y$ (or $1/X$ and $1/Y$) are independently normally distributed with mean zero and variance $\sigma^2,$ then $X/Y$ has a Cauchy distribution. It is the purpose of this note to show that the converse statement is not true. That is, the fact that the ratio of two independent, identically distributed, random variables $X$ and $Y$ follows a Cauchy distribution is not sufficient to imply that $X$ and $Y$ (or $1/X$ and $1/Y$) are normally distributed. This will be shown by exhibiting several counterexamples.