ABSTRACT
The α-symmetric multivariate distribution has been introduced in Section 1.1, as a natural generalization of an i.i.d. sample of symmetric stable law (1.8). In fact, the spherical distribution is a special case of the α-symmetric distribution with α = 2. The progress of α-symmetric distributions for other cases of α has been relatively slow, perhaps due to the fact that the analysis with characteristic functions heavily involves calculus of complex variables which is not a favourite tool among statisticians. This may change, however, especially when statisticians realize that distributions encountered in sciences and social sciences need not possess nice explicit p.d.f.s nor finite moments, and that it is still possible to do statistical analysis with such distributions. The excellent exposition on one-dimensional stable distributions by Zolotarev (1985) may well attract the attention of statisticians in this direction. In this last chapter, we shall present the general properties of α-symmetric distributions and consider some special cases of α and n. We emphasize that the problems listed at the end will form an integral part of this chapter and will be referred to from time to time. The results presented here are mainly taken from Cambanis, Keener and Simons (1983) and Kuritsyn and Shestakov (1984). Some unpublished results of K.W. Ng are also incorporated in the presentation.
