The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. It has parameter, α, which is defined over the range 0 < α ≤ 2, and where the case α = 2 is equivalent to the multivariate normal distribution. It has an additional skew parameter that allows for non-symmetric distributions, where the multivariate normal distribution is symmetric.
Let be the unit sphere in . A random vector, , has a multivariate stable distribution - denoted as -, if the joint characteristic function of is
where 0 < α < 2, and for
This is essentially the result of Feldheim, that any stable random vector can be characterized by a spectral measure (a finite measure on ) and a shift vector .
Parametrization using projections
Another way to describe a stable random vector is in terms of projections. For any vector , the projection is univariate stable with some skewness , scale and some shift . The notation is used if X is stable with
for every . This is called the projection parameterization.
The spectral measure determines the projection parameter functions by:
There are special cases where the multivariate characteristic function takes a simpler form. Define the characteristic function of a stable marginal as
Isotropic multivariate stable distribution
The characteristic function is
The spectral measure is continuous and uniform, leading to radial/isotropic symmetry.
For the multinormal case , this corresponds to independent components, but so is not the case when . Isotropy is a special case of ellipticity (see the next paragraph) – just take to be a multiple of the identity matrix.
Elliptically contoured multivariate stable distribution
The elliptically contoured multivariate stable distribution is a special symmetric case of the multivariate stable distribution.
If X is α-stable and elliptically contoured, then it has joint characteristic function
for some shift vector (equal to the mean when it exists) and some positive definite matrix (akin to a correlation matrix, although the usual definition of correlation fails to be meaningful).
Note the relation to characteristic function of the multivariate normal distribution: obtained when α = 2.
The marginals are independent with , then the
characteristic function is
Observe that when α = 2 this reduces again to the multivariate normal; note that the iid case and the isotropic case do not coincide when α < 2.
Independent components is a special case of discrete spectral measure (see next paragraph), with the spectral measure supported by the standard unit vectors.
If the spectral measure is discrete with mass at
the characteristic function is
If is d-dimensional, A is an m x d matrix, and
then AX + b is m-dimensional -stable with scale function skewness function and location function
Inference in the independent component model
Recently it was shown how to compute inference in closed-form in a linear model (or equivalently a factor analysis model), involving independent component models.
More specifically, let be a set of i.i.d. unobserved univariate drawn from a stable distribution. Given a known linear relation matrix A of size , the observation are assumed to be distributed as a convolution of the hidden factors . . The inference task is to compute the most probable , given the linear relation matrix A and the observations . This task can be computed in closed-form in O(n3).
^D. Bickson and C. Guestrin. Inference in linear models with multivariate heavy-tails. In Neural Information Processing Systems (NIPS) 2010, Vancouver, Canada, Dec. 2010. https://www.cs.cmu.edu/~bickson/stable/