On asymptotic properties of multidimensional α-stable densities

Univariate stable distributions arose within the context of the central limit theorem as limit laws for sums of i.i.d. random variables. Except for the gaussian laws, all the stable distributions are heavy-tailed. This explains why stable distribution models are very attractive for statisticians. The fact that stable laws are limiting says a lot about possible genesis of the corresponding random variables. Analytical properties of univariate stable distributions are well studied (see e.g. [31] and [5, 10, 11, 29]). The notion of stability can be carried over to higher dimensions in various ways. The simplest way is the following. Let η, η, η, . . . be i.i.d. random vectors. It is said that the distribution of η is α-stable in R if for any n > 1 there exists a ∈ R and a number α ∈ (0, 2] such that