A Decomposition Theorem for Vector Variables with a Linear Structure

A vector variable X is said to have a linear structure if it can be written as X=AY where A is a matrix and Y is a vector of independent random variables called structural variables. In earlier papers the conditions under which a vector random variable admits different structural representations have been studied. It is shown, among other results, that complete non-uniqueness, in some sense, of the linear structure characterizes a multivariate normal variable. In the present paper we prove a general decomposition theorem which states that any vector variable X with a linear structure can be expressed as the sum X 1 + X 2 of two independent vector variables X 1 , X 2 of which X 1 is non-normal and has a unique linear structure, and X 2 is multivariate normal variable with a nonunique linear structure.