AN UNEXPECTED ENCOUNTER WITH CAUCHY AND LEVY

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 >= 3. Let X = (X-1,...,X-m) and Y = (Y-1 ,...,Y-m) be i.i.d. N(0, Sigma), where Sigma = {sigma(ij)} >= 0 is an m x m and arbitrary covariance matrix with sigma(jj) > 0 for all 1 <= j <= m. Then Z = Sigma(m)(j=1) w(j) X-j/Y-j similar to Cauchy (0, 1), as long as (w) over right arrow = (w(1),...,w(m)) is independent of (X, Y), w(j) >= 0, j = 1,..., m, and Sigma(m)(j=1) w(j) = 1. In this note, we present an elementary proof of this conjecture for any m >= 2 by linking Z to a geometric characterization of 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 Sigma(m)(i=1) Sigma(m)(j=1) w(i)w(j)sigma(ij)/XiXj similar to Levy (0,1) This generalizes the super Cauchy phenomenon that the average of m i.i.d. standard Levy variables (i.e., inverse chi-squared variables with one degree of freedom) has the same distribution as that of a single standard Levy variable multiplied by m (which is obtained by taking w(j) = 1/m and Sigma to be the identity matrix).