On characterization of gamma and multivariate normal distributions by solving some functional equations in vector variables

AbstractThe main aim of this paper is to solve the functional equations ∑i=1kCαihi(ti)+∑j=1rBαjgi∑i=1kA′ijti=dα, α=1,2,…,q, and k > 1, in vector variables t1,…,tk satisfying the condition |ti| = (Σj = 1m tij2)1/2 < δ for all i, where Cαi, Bαj and Aij are given square matrices and dα is a given vector. The elements of the vector functions hi and gj are unknown continuous functions in vector variables. The study of the other form ∑j=1rAijψ(Bj)=difo i=1,2…,q is postponed at present and will be published in a later communication. The applications of the first set of functional equations are given in characterizing the gamma and the multivariate normal distributions at the end of the paper.In order to solve the required set of functional equations, the certain product of two matrices, as defined by Khatri and Rao (1968b), is extended. Let A = (A1,…, Ar) and B = (B1,…, Br). Then the extended product AB is defined by the partitioned matrix (A1⊗B1,…Ar⊗Br), where ⊗ denotes the Kronecker product. Some interesting properties of this extended product along with some useful results in matrix algebra are established