A note on multivariate linear exponential distributions
D. C. Doss · Canadian Journal of Statistics · 1974
Abstract A distribution admitting the moment generating function (mgf) is a member of one and only one linear exponential family Pμ = {Pω:ω ϵ Ωμ} with dPω(x) = {eω'x/f(ω)} dμ(x). Factorization of the mgf of a member of a linear exponential family into a finite number of mgf's of linear exponential distributions, implies such a factorization for all members of the family. Consequently, infinite divisibility of the mgf of a member of a linear exponential family implies infinite divisibility for all members of the family.