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As a sub-class of multivariate $\alpha$-stable distributions, the elliptically contoured $\alpha$-stable or sub-Gaussian $\alpha$-stable (SGS) seemingly is simpler to evaluate and more tractable than the multivariate $\alpha$-stable distributions. Although the density function of the SGS distribution has not closed form, but it characteristic function does. Suppose that ${\color{blue}{{\boldsymbol{X}}=(X_1,\cdots,X_d)^{\top}}}$ follows SGS distribution, the characteristic function of SGS distribution is given by

$${\color{blue}{\exp\Bigl[- i \langle {\boldsymbol{t}}, {\boldsymbol{X}} \rangle\Bigr]=\exp\Bigl[- \bigl( {\boldsymbol{t}}^{\top} \Sigma {\boldsymbol{t}} \bigr)^{\frac{\alpha}{2}} + i \langle {\boldsymbol{t}}, {\boldsymbol{\mu}} \rangle \Bigr],}}$$

where $\Sigma$ is a $d \times d$ positive definite dispersion matrix, ${\boldsymbol{\mu}}$ is the location parameter, and symbol $\langle . , .\rangle$ accounts for the inner product in $\mathbb{R}^{d}$. Each SGS distribution can be regarded as a Gaussian-Scale Mixture (GSM) model. Let ${\boldsymbol{G}}$ follow a multivariate Gaussian distribution with covariance matrix $\Sigma$ and mean vector ${\bf{0}}$ of appropriate size. We can write $${\color{blue}{{\boldsymbol{X}}=\sqrt{2P} {\boldsymbol{G}} + {\boldsymbol{\mu}},}}$$ where $P$ follows a positive $\alpha$-stable distribution, that is, ${\color{blue}{P\sim S(\alpha/2, 1, (\cos(\pi\alpha/4))^{2/\alpha}, 0)}}$. Here, positive $\alpha$-stable distribution plays the role of mixing distribution. The GSM models can produce a symmetric distribution with tails heavier than the Gaussian model.