Darmois–Skitovich theorem

The Darmois–Skitovich theorem  is one of the most famous characterization theorems of mathematical statistics. It characterizes the normal distribution (the Gaussian distribution) by the independence of two linear forms from independent random variables. This theorem was proved independently by G. Darmois and V. P. Skitovich in 1953.

Formulation

Let   be independent random variables. Let   be nonzero constants. If the linear forms and are independent then all random variables have normal distributions (Gaussian distributions).

History

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]

Information sources

References

  1. "О теорем Дармуа-Скитовича" (PDF). www.apmath.spbu.ru (in Russian).
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