Ratios and Cauchy Distribution
نویسنده
چکیده
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 [PM16], 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, . . . , wm) be a vector of non-negative numbers with ∑m j=1 wj = 1. It was conjectured in [DX16], and proved in [PM16], that the random variable
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