Ratios and Cauchy Distribution

2016/02/26 by Pillai, Natesh S.
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1602.08181

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 Σ be a m× m positive semi-definite matrix. Let X,Y ∼ N(0,Σ) be independent vectors. Let w=(w₁, …, w_m) be a vector of non-negative numbers with ∑_j=1^m w_j = 1. The author proved recently that the random variable Z = ∑_j=1^m w_j(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.

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