Constraints on distributions imposed by properties of linear forms

Institute fur Angewandte Mathematik, Universität Bonn, Interdisziplinares Zentrum für Komplexe Systeme, Meckenheimer Allee 176, 53115 Bonn, Germany; db@izks.uni-bonn.de.

Received: 22 April 2002
Revised: 17 March 2003

Abstract

Let (X₁,Y₁),...,(X_m,Y_m) be m independent identically distributed bivariate vectors and L₁ = β₁X₁ + ... + β_mX_m, L₂ = β₁X₁ + ... + β_mX_m are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of L₁ and L₂ imply the same property for X₁ and Y₁, and under what conditions does the independence of L₁ and L₂ entail independence of X₁ and Y₁? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.