An unexpected encounter with Cauchy and Levy

Natesh Pillai. Dept of Statistics, Harvard University

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 (2014) 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,Σ), where Σ={σ_ij}≥0 is an m×m arbitrary covariance matrix with σ_jj>0 for all 1≤j≤m.
Then Z=∑_j   w_j   X_j  Y_j ∼Cauchy(0,1), as long as w=(w_1,...,w_m) is independent of (X,Y), w_j≥0,j=1,...,m, and ∑_j w_j=1.
In this talk, 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 (1969).

Joint work with Xiao-Li Meng.