Transformations preserving normality and Wishart-ness

Let Rn-p, (n), Gl(p) and +(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[reverse not equivalent]Rp-1 preserving the multivariate normality of Np([mu], [Sigma]) for fixed [mu]=[mu]1, [mu]2 ([mu]1[not equal to][mu]2) and for all [Sigma][set membership, variant]+(p) is of the form g(x)=Ax+b a.e. for some (A, b)[set membership, variant]Gl(p)-Rp. Second, a BBT g on Rn-p preserving the form for certain 's and all [Sigma][set membership, variant]+(p) is shown to be of the form g(x)=QxA+E a.e. for some (Q, A, E)[set membership, variant](n)-Gl(p)-Rn-p. Third, a BBT h on +(p) preserving the Wishart-ness of Wp([Sigma], m) (m>=p) for all [Sigma][set membership, variant]+(p) is shown to be of the form h(w)=A'wA a.e. for some A[set membership, variant]Gl(p). Fourth, a BBT k(x, w)=(k1(x, w), k2(x, w)) on Rn-p-+(p) which preserves the form of for certain 's and all [Sigma][set membership, variant]+(p) is shown to be of the form k(x, w)=(QxA+E, A'wA) a.e. for some (Q, A, E)[set membership, variant](n)-Gl(p)-Rn-p.