Comparison of tail probabilities of strictly semistable/stable random vectors and their symmetrized counterparts with application

It is shown that the tail probabilities of a strictly (r, a)-semistable (0 < r < 1, 0 < α < 2, α ≠ 1) Banach space valued random vector X and its symmetrized counterpart are ”uniformly” comparable in the sense that the constants appearing in the inequalities depend only on r and α (and not on X or the Banach space). Using this and some other known facts, several corollaries related to the moment inequalities of the random vector X and its symmetrized counterpart are obtained. The corresponding results for strictly α-stable Banach space valued random vectors, α ≠ 1, are also derived and discussed.