When are Two Special Linear Forms of Independent Random Vectors Identically Distributed?

Abstract

Let X _i , i=1, …, n be independent P _i-dimensional random vectors. Consider two linear functions

$$ \text{L}_\text{1} \text{ = A}_\text{1} \text{X}_\text{1} \text{ + \ldots + A}_n \text{X}_{n,} $$

$$ \text{L}_\text{2} \text{ = B}_\text{1} \text{X}_\text{1} \text{ + \ldots + B}_n \text{X}_{n,} $$

where A _i , B _i are (mxP _i ) matrices. In the paper the condition of identically distribution of and L₁ and L₂ is studied. The results are applied to a characterization of the normal law and to the problem of characterization of probability distributions of random vectors given the joint distribution of linear functions of them.

The results provide a generalization of those concerning characterization of probability laws throught properties of linear functions.