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Let ( X 1 ,Y 1 ),...,(X m ,Y m ) be m independent identically distributed bivariate vectors and L 1 = β 1 X 1 + ... + β m X m , L 2 = β 1 X 1 + ... + β m X m are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of L 1 and L 2 imply the same property for X 1 and Y 1 , and under what conditions does the independence of L 1 and L 2 entail independence of X 1 and Y 1 ? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.

Constraints on distributions imposed by properties of linear forms