An unexpected encounter with Cauchy and Lévy

Abstract

The Cauchy distribution is usually presented as a mathematical curiosity, an exception to the Law of Large Numbers, or even as an “Evil” distribution in some introductory courses. It therefore surprised us when Drton and Xiao [Bernoulli 22 (2016) 38–59] proved the following result for $m=2$ and conjectured it for $m\ge3$. Let $X=(X_{1},\ldots,X_{m})$ and $Y=(Y_{1},\ldots,Y_{m})$ be i.i.d. $\mathrm{N}(0,\Sigma)$, where $\Sigma=\{\sigma_{ij}\}\ge0$ is an $m\times m$ and arbitrary covariance matrix with $\sigma_{jj}>0$ for all $1\leq j\leq m$. Then

\[Z=\sum_{j=1}^{m}w_{j}{\frac{X_{j}}{Y_{j}}}\sim\operatorname{Cauchy}(0,1),\] as long as $\vec{w}=(w_{1},\ldots,w_{m})$ is independent of $(X,Y)$, $w_{j}\ge0,j=1,\ldots,m$, and $\sum_{j=1}^{m}w_{j}=1$. In this note, we present an elementary proof of this conjecture for any $m\geq2$ by linking $Z$ to a geometric characterization of $\operatorname{Cauchy}(0,1)$ given in Willams [Ann. Math. Stat. 40 (1969) 1083–1085]. This general result is essential to the large sample behavior of Wald tests in many applications such as factor models and contingency tables. It also leads to other unexpected results such as

\[\sum_{i=1}^{m}\sum_{j=1}^{m}\frac{w_{i}w_{j}\sigma_{ij}}{X_{i}X_{j}}\sim \mbox{Lévy}(0,1).\] This generalizes the “super Cauchy phenomenon” that the average of $m$ i.i.d. standard Lévy variables (i.e., inverse chi-squared variables with one degree of freedom) has the same distribution as that of a single standard Lévy variable multiplied by $m$ (which is obtained by taking $w_{j}=1/m$ and $\Sigma$ to be the identity matrix).