Multivariate stable distribution

The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function.Heatmap showing a multivariate (bivariate) independent stable distribution with α = 2. The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function. 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 S {displaystyle mathbb {S} } be the unit sphere in R d : S = { u ∈ R d : | u | = 1 } {displaystyle mathbb {R} ^{d}colon mathbb {S} ={uin mathbb {R} ^{d}colon |u|=1}} . A random vector, X {displaystyle X} , has a multivariate stable distribution - denoted as X ∼ S ( α , Λ , δ ) {displaystyle Xsim S(alpha ,Lambda ,delta )} -, if the joint characteristic function of X {displaystyle X} is where 0 < α < 2, and for y ∈ R {displaystyle yin mathbb {R} } This is essentially the result of Feldheim, that any stable random vector can be characterized by a spectral measure Λ {displaystyle Lambda } (a finite measure on S {displaystyle mathbb {S} } ) and a shift vector δ ∈ R d {displaystyle delta in mathbb {R} ^{d}} . Another way to describe a stable random vector is in terms of projections. For any vector u {displaystyle u} , the projection u T X {displaystyle u^{T}X} is univariate α − {displaystyle alpha -} stable with some skewness β ( u ) {displaystyle eta (u)} , scale γ ( u ) {displaystyle gamma (u)} and some shift δ ( u ) {displaystyle delta (u)} . The notation X ∼ S ( α , β ( ⋅ ) , γ ( ⋅ ) , δ ( ⋅ ) ) {displaystyle Xsim S(alpha ,eta (cdot ),gamma (cdot ),delta (cdot ))} is used if X is stable with u T X ∼ s ( α , β ( ⋅ ) , γ ( ⋅ ) , δ ( ⋅ ) ) {displaystyle u^{T}Xsim s(alpha ,eta (cdot ),gamma (cdot ),delta (cdot ))} for every u ∈ R d {displaystyle uin mathbb {R} ^{d}} . This is called the projection parameterization.