Transformations preserving normality and Wishart-ness

AbstractLet Rn×p, O(n), Gl(p) and S+(p) denote respectively the set of n×p matrices, the set of n×n orthogonal matrices, the set of p×p nonsingular matrices and the set of p × p positive definite matrices. In this paper, it is first shown that a bijective and bimeasurable transformation (BBT) g on Rp≡Rp×1 preserving the multivariate normality of Np(μ, Σ) for fixed μ=μ1, μ2 (μ1≠μ2) and for all Σ∈S+(p) is of the form g(x)=Ax+b a.e. for some (A, b)∈Gl(p)×Rp. Second, a BBT g on Rn×p preserving the form Nn×p(, I⊗Σ) for certain 's and all Σ∈S+(p) is shown to be of the form g(x)=QxA+E a.e. for some (Q, A, E)∈O(n)×Gl(p)×Rn×p. Third, a BBT h on S+(p) preserving the Wishart-ness of Wp(Σ, m) (m≥p) for all Σ∈S+(p) is shown to be of the form h(w)=A′wA a.e. for some A∈Gl(p). Fourth, a BBT k(x, w)=(k1(x, w), k2(x, w)) on Rn×p×S+(p) which preserves the form of Nn×p(, I⊗Σ)×Wp(Σ, m) for certain 's and all Σ∈S+(p) is shown to be of the form k(x, w)=(QxA+E, A′wA) a.e. for some (Q, A, E)∈O(n)×Gl(p)×Rn×p