##
**On the distributions of sums of symmetric random variables and vectors.**
*(English)*
Zbl 0624.60025

A probability distribution F on \({\mathbb{R}}\) is said to be balanced if it either has mean zero or no mean. 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.

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.

Reviewer: S.T.Rachev

### MSC:

60E05 | Probability distributions: general theory |

60E10 | Characteristic functions; other transforms |