SOME CHARACTERIZATIONS OF THE CAUCHY DISTRIBUTION

If X and Y are independent standard Cauchy random variables then (i) Y and (X+Y)/(1-Xu) are independent, (ii) X and (X + Y)/(1 -XU) are identically distributed, and (iii) X and 2X/(1-X²) are identically distributed. Each of these three properties is shown to characterize the Cauchy distribution among absolutely continuous distributions. Some related uniform characterizations are discussed.