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Independence of linear forms with random coefficients. (English) Zbl 1105.62052

Summary: We extend the classical Darmois-Skitovich theorem to the case where the linear forms \(L_{r1} = U_1 X_1 + \cdots + U_n X_n\) and \(L_{r2} = U_{n+1} X_1 + \cdots + U_{2n} X_n\) have random coefficients \(U_1,\ldots,U_{2n}\). Under minimal restrictions on the random coefficients we completely describe the distributions of the independent random variables \(X_1,\ldots,X_n\) and \(U_1,\ldots,U_{2n}\) such that the linear forms \(L_{r1}\) and \(L_{r2}\) are independent.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E10 Characterization and structure theory of statistical distributions
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