Sample Means of Independent Standard Cauchy Random Variables Are Standard Cauchy: A New Approach

Abstract

A remarkable property of the Cauchy distribution is that the sample mean of standard Cauchy random variables itself has a standard Cauchy distribution. This result can be shown using characteristic functions, convolution integrals, or multidimensional change of variables (Jacobians). A new method of proof based on the periodicity properties of the tangent function is presented.