is there an analytic formula for fractional moments of multivariate normal distribution? E(∏ki=1xνii)=?E(∏i=1kxiνi)=? where X=(x1,…,xk)∼Nk(μ,Σ)X=(x1,…,xk)∼Nk(μ,Σ), νi∈Rνi∈R and νi>0νi>0. I know there is one for univariate normal distributions, but can’t find one for the multivariate case. Many thanks in advance.
The formula for univariate case I know of is E[|x−μ|ν]=σν2ν/2Γ(ν+12)π√E[|x−μ|ν]=σν2ν/2Γ(ν+12)π. I understand this is about the central absulate moment, but I am more interested in E(∏ki=1xνii)E(∏i=1kxiνi) or E(∏ki=1|xi|νi)E(∏i=1k|xi|νi).