Elliptical Distributions. Vadym Omelchenko. Examples of the Elliptical Distributions. Normal Distribution Laplace Distribution t-Student Distribution Cauchy Distribution Logistic Distribution Symmetric Stable Laws. Examples of the Elliptical Distributions.
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Thefurtherρfromzerothe more evidentellipticity of the map, when observing it from above. When ρ=0 then the map has the spherical form.
and Σ are given by
If X has an elliptical distribution, we write X ̴̴̴ where is called characteristic generator of X and hence, the characteristic generator of the multivariate normal is given by
For some non-negative function called density generator and for some constant called normalizing constant.
Corollary. Let X ̴ . Then ̴
Hence marginal distributions of elliptical distributions are elliptical distributions.
guarantees that is a density generator.
Suppose X ̴ is the vector of ones with dimension n, and