A decomposition theorem for vector variables with a linear structure

Abstract

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 X₁ + X₂ of two independent vector variables X₁, X₂ of which X₁ is non-normal and has a unique linear structure, and X₂ is multivariate normal variable with a nonunique linear structure.