Darmois–Skitovich theorem
The Darmois–Skitovich theorem is one of the most famous characterization theorems of mathematical statistics. It characterizes the normal 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.History
The Darmois-Skitovich theorem is a generalization of the Kac-Bernstein theorem in which the normal 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
