Kac–Bernstein theorem

The Kac–Bernstein theorem is one of the first characterization theorems of mathematical statistics. If the random variables ${\displaystyle \xi }$  and ${\displaystyle \eta }$  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.