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.
Definition
Let be the unit sphere in . A random vector, , has a multivariate stable distribution - denoted as -, if the joint characteristic function of is[1]
where 0 < α < 2, and for
This is essentially the result of Feldheim,[2] 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:
Special cases
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.[3]
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.
Independent components
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.
Heatmap showing a multivariate (bivariate) independent stable distribution with α = 1
Heatmap showing a multivariate (bivariate) independent stable distribution with α = 2
Discrete
If the spectral measure is discrete with mass at
the characteristic function is
Linear properties
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[4] 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).
An application for this construction is multiuser detection with stable, non-Gaussian noise.
^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/