On the Laws of Cauchy and Gauss

Abstract

Let $x$ and $y$ be two independent normal variates each distributed with zero mean and a common variance; it is then well-known that the quotient $x/y$ follows the Cauchy law distributed symmetrically about the origin. Now the question that naturally arises is whether we can obtain a characterization of the normal distribution by this property of the quotient. This converse problem can be more precisely formulated as follows: Let $x$ and $y$ be two independently and identically distributed random variables having a common distribution function $F(x)$. Let the quotient $w = x/y$ follow the Cauchy law distributed symmetrically about the origin $w = 0$. Then the question is whether $F(x)$ is normal. But this converse is not true in general. The author [1] has recently constructed a very simple example of a non-normal distribution where the quotient $x/y$ follows the Cauchy law. Steck [7] has also given some examples of non-normal distributions with this property of the quotient. In the present paper we shall first derive some interesting general properties possessed by the class of distribution laws $F(x)$ [Section 2]. In Section 3 we deduce a characterization of the normal distribution under some conditions on the distribution function $F(x)$. Finally in Section 4 we construct an example of a non-normal distribution function $F(x)$ having finite moments of all orders where the quotient $x/y$ follows the Cauchy law. The method of proof is essentially based on the applications of Fourier transforms of distribution functions. For the proof of Theorem 3.1 we require somewhat deeper results in the theory of analytic functions.