On Symmetric Stable Random Variables and Matrix Transposition

In a companion paper, the authors obtained some Fubini type identities in law for quadratic functionals of Brownian motion, and, more generally, for certain functionals of symmetric stable processes, the function: x -* x2 then being replaced by: x e IxIa. In this paper, discrete analogues of such identities in law, which involve a sequence of independent standard symmetric stable r.v.'s of index a, are presented. It is then shown that such identities in law characterize the symmetric a-stable distribution. Some related characterization results, either for some finite or infinite dimensional r.v.'s are also presented.