Darmois–Skitovich theorem

Let ${\displaystyle \xi _{j},j=1,2,\ldots ,n,n\geq 2}$  be independent random variables. Let ${\displaystyle \alpha _{j},\beta _{j}}$  be nonzero constants. If the linear forms ${\displaystyle L_{1}=\alpha _{1}\xi _{1}+\cdots +\alpha _{n}\xi _{n}}$ and ${\displaystyle L_{2}=\beta _{1}\xi _{1}+\cdots +\beta _{n}\xi _{n}}$ are independent then all random variables ${\displaystyle \xi _{j}}$ have normal distributions (Gaussian distributions).

The Darmois-Skitovich theorem is a generalization of the Kac-Bernstein theorem in which the normal distribution (the Gauss distribution) is characterized by the independence of the sum and the difference of two independent random variables. For a history of proving the theorem by V.P.Skitovich, see the article [1]

• Darmois, G. (1953). Analyse generale des liaisons stochastiques. Rev.Inst.Intern.Stat (21): 2—8.
• Skitivic, V. P. (1953). "On a property of the normal distribution." Dokl. Akad. Nauk SSSR (N.S.) (89): 217—219 (in Russian).
• A. M. Kagan, Yu. V. Linnik, and C. R. Rao, Characterization Problems in Mathematical Statistics, Wiley, New York (1973).