Independence of linear forms with random coefficients

We extend the classical Darmois-Skitovich theorem to the case where the linear forms L-r1 = U1X1 +center dot center dot center dot+ UnXn and L-r2 = Un+1X1+center dot center dot center dot+U2nXn have random coefficients U-1,...,U-2n. Under minimal restrictions on the random coefficients we completely describe the distributions of the independent random variables X-1,...,X-n and U-1,...,U-2n such that the linear forms L-r1 and L-r2 are independent.