An overview of multivariate stable distributions

A d-dimensional α-stable random vector is determined by a spectral measure Γ (a finite Borel measure on Sd=unit sphere in R) and a shift vector μ ∈ R, e.g. Samorodnitsky and Taqqu (1994). The notation X ∼ Sα,d(Γ, μ) will be used to denote such a stable random vector. Until recently, there has been little understanding of what multivariate stable distributions look like, nor any methods for working with them. In this paper we review recent work that makes it possible to use multivariate stable models for practical problems. Our goal here is make these results available to people who want to understand and use multivariate stable densities, so we state results without proof; references are given for those who are interested. Below we summarize some basic facts about multivariate stable distributions. In Section 2 we describe the class of stable random variables that correspond to discrete spectral measures with a finite number of point masses and show that they are dense in the set of all stable measures. Then in Section 3, formulas are given for computing multivariate stable densities and examples of such computations are given. A method of generating stable random vectors is given in Section 4. Section 5 is concerned with the problem of estimating a spectral measure from a data set, and the last section discusses the problem of identifying the type of a stable process by looking at the spectral measure associated with finite dimensional distributions of the process. The characteristic function of X ∼ Sα,d(Γ, μ) is φX(t) = E exp{i < X, t >} = exp(−IX(t) + i < μ, t >), (1)