Let (X1,Y1),...,(Xm,Ym) be m independent identically
distributed bivariate vectors
and
L1 = β1X1 + ... + βmXm, L2 = β1X1 + ... + βmXm
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.