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 and are independent and normally distributed, 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. This theorem was proved independently by Polish-American mathematician Mark Kac and Soviet mathematician Sergei Bernstein.Formulation
Let and are independent random variables. If and are independent then and have normal distributions.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.
