A characterization of the multivariate normal distribution

Let Xj (j = 1,...,n) be i.i.d. random variables, and let Y' = (Y1,...,Ym) and X' = (X1,...,Xn) be independently distributed, and A = (ajk) be an n - n random coefficient matrix with ajk = ajk(Y) for j, K = 1,...,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist. 41 (1970)] have shown U ~ N(O,I) if and only if X ~ N(O,I). provided that certain conditions defined in terms of the ajk are satisfied. The task of this paper is to delete the identical assumption on X1,...,Xn and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.