Multivariate stable distribution

Let ${\displaystyle \mathbb {S} }$ be the unit sphere in ${\displaystyle \mathbb {R} ^{d}\colon \mathbb {S} =\{u\in \mathbb {R} ^{d}\colon |u|=1\}}$. A random vector, ${\displaystyle X}$, has a multivariate stable distribution - denoted as ${\displaystyle X\sim S(\alpha ,\Lambda ,\delta )}$ -, if the joint characteristic function of ${\displaystyle X}$ is[1]

${\displaystyle \operatorname {E} \exp(iu^{T}X)=\exp \left\{-\int \limits _{s\in \mathbb {S} }\left\{|u^{T}s|^{\alpha }+i\nu (u^{T}s,\alpha )\right\}\,\Lambda (ds)+iu^{T}\delta \right\}}$

where 0 < α < 2, and for ${\displaystyle y\in \mathbb {R} }$

${\displaystyle \nu (y,\alpha )={\begin{cases}-\mathbf {sign} (y)\tan(\pi \alpha /2)|y|^{\alpha }&\alpha \neq 1,\\(2/\pi )y\ln |y|&\alpha =1.\end{cases}}}$

This is essentially the result of Feldheim,[2] that any stable random vector can be characterized by a spectral measure ${\displaystyle \Lambda }$ (a finite measure on ${\displaystyle \mathbb {S} }$) and a shift vector ${\displaystyle \delta \in \mathbb {R} ^{d}}$.

Another way to describe a stable random vector is in terms of projections. For any vector ${\displaystyle u}$, the projection ${\displaystyle u^{T}X}$ is univariate ${\displaystyle \alpha -}$stable with some skewness ${\displaystyle \beta (u)}$, scale ${\displaystyle \gamma (u)}$ and some shift ${\displaystyle \delta (u)}$. The notation ${\displaystyle X\sim S(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))}$ is used if X is stable with ${\displaystyle u^{T}X\sim s(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))}$ for every ${\displaystyle u\in \mathbb {R} ^{d}}$. This is called the projection parameterization.

The spectral measure determines the projection parameter functions by:

${\displaystyle \gamma (u)={\Bigl (}\int _{s\in \mathbb {S} }|u^{T}s|^{\alpha }\Lambda (ds){\Bigr )}^{1/\alpha }}$

${\displaystyle \beta (u)=\int _{s\in \mathbb {S} }|u^{T}s|^{\alpha }\mathbf {sign} (u^{T}s)\Lambda (ds)/\gamma (u)^{\alpha }}$

${\displaystyle \delta (u)={\begin{cases}u^{T}\delta &\alpha \neq 1\\u^{T}\delta -\int _{s\in \mathbb {S} }{\tfrac {\pi }{2}}u^{T}s\ln |u^{T}s|\Lambda (ds)&\alpha =1\end{cases}}}$

There are special cases where the multivariate characteristic function takes a simpler form. Define the characteristic function of a stable marginal as

${\displaystyle \omega (y|\alpha ,\beta )={\begin{cases}|y|^{\alpha }\left[1-i\beta (\tan {\tfrac {\pi \alpha }{2}})\mathbf {sign} (y)\right]&\alpha \neq 1\\|y|\left[1+i\beta {\tfrac {2}{\pi }}\mathbf {sign} (y)\ln |y|\right]&\alpha =1\end{cases}}}$

Independent components

The marginals are independent with ${\displaystyle X_{j}\sim S(\alpha ,\beta _{j},\gamma _{j},\delta _{j})}$, then the characteristic function is

${\displaystyle E\exp(iu^{T}X)=\exp \left\{-\sum _{j=1}^{m}\omega (u_{j}|\alpha ,\beta _{j})\gamma _{j}^{\alpha }+iu^{T}\delta )\right\}}$

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

If ${\displaystyle X\sim S(\alpha ,\beta (\cdot ),\gamma (\cdot ),\delta (\cdot ))}$ is d-dimensional, A is an m x d matrix, and ${\displaystyle b\in \mathbb {R} ^{m},}$ then AX + b is m-dimensional ${\displaystyle \alpha }$-stable with scale function ${\displaystyle \gamma (A^{T}\cdot ),}$ skewness function ${\displaystyle \beta (A^{T}\cdot ),}$ and location function ${\displaystyle \delta (A^{T}\cdot )+b^{T}.}$

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 ${\displaystyle X_{i}\sim S(\alpha ,\beta _{x_{i}},\gamma _{x_{i}},\delta _{x_{i}}),i=1,\ldots ,n}$ be a set of i.i.d. unobserved univariate drawn from a stable distribution. Given a known linear relation matrix A of size ${\displaystyle n\times n}$, the observation ${\displaystyle Y_{i}=\sum _{i=1}^{n}A_{ij}X_{j}}$ are assumed to be distributed as a convolution of the hidden factors ${\displaystyle X_{i}}$. ${\displaystyle Y_{i}=S(\alpha ,\beta _{y_{i}},\gamma _{y_{i}},\delta _{y_{i}})}$. The inference task is to compute the most probable ${\displaystyle X_{i}}$, given the linear relation matrix A and the observations ${\displaystyle Y_{i}}$. This task can be computed in closed-form in O(n³).

An application for this construction is multiuser detection with stable, non-Gaussian noise.

• Multivariate Cauchy distribution
• Multivariate normal distribution

• Mark Veillette's stable distribution matlab package http://www.mathworks.com/matlabcentral/fileexchange/37514
• The plots in this page where plotted using Danny Bickson's inference in linear-stable model Matlab package: https://www.cs.cmu.edu/~bickson/stable