Kac–Bernstein theorem
The Kac–Bernstein theorem  is one of the first characterization theorems of mathematical statistics. It is easy to see that if the random variables formula_1  and formula_2  are independent and normally distributed with the same variance, then their sum and difference are also independent. The Kac–Bernstein theorem states that the independence of the sum and difference of two independent random variables characterizes the normal distribution (the Gauss distribution). This theorem was proved independently by Polish-American mathematician Mark Kac and Soviet mathematician Sergei Bernstein. Formulation. Let formula_1  and formula_2  are independent random variables. If formula_5  and  formula_6  are independent then formula_1  and formula_2  have normal distributions (the Gaussian distribution). Generalization. A generalization of the Kac–Bernstein theorem is the Darmois–Skitovich theorem, in which instead of sum and difference linear forms from "n" independent random variables are considered.