Functional equations and characterization of probability laws through linear functions of random variables

General functional equations of the type [summation operator][phi]i(Ai't+Bi'u)=Ca(ut)+Db(tu)+Pk(t,u) and [Sigma][phi]i(Ci't) = Pk(t) have been solved, where Pk represents a polynomial of degree k in all the arguments, Cn(u [short parallel] t), a polynomial of degree a in u given t, and Db(t [short parallel] u), a polynomial of degrce b in t given u. The results are applied in characterizing the multivariate normal variable by nonuniqueness of linear structure, independence of sets of linear functions, and constancy of regression of one set on another set of linear functions. The problem of characterization of probability distributions of individual random variables (which may be vectors), given the joint distribution of a relatively few linear functions of all the variables, has been studied. The results provide a generalization of all previous work on the characterization of probability laws of vector random variables through linear functions.