On the distributions of sums of symmetric random variables and vectors (Q578720)

A probability distribution F on \({\mathbb{R}}\) is said to be balanced if it either has mean zero or no mean. \textit{G. Simons} [An interesting application of Fatou's lemma. Am. Stat. 30, 146 (1976)] showed that a necessary condition for F to be the distribution of a sum \(X+Y\) of dependent random variables X, Y symmetric about zero, is that F is balanced. The authors of the present paper proved that this condition is also sufficient. Considering the n-dimensional case they showed that there exist spherically symmetric random vectors X and Y whose sum has n-dimensional distribution F if and only if F is balanced in the multidimensional sense, i.e. all the one-dimensional distributions obtained by projecting F onto lines through the origin are balanced. As an example of their construction they show that, for every positive integer n, there exist n-dimensional Cauchy random vectors X and Y, spherically symmetric about the origin, such that the sum \(X+Y\) has an n- dimensional Cauchy distribution which is spherically symmetric about a point other than the origin.