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 ${\displaystyle \xi }$  and ${\displaystyle \eta }$  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 (the Gauss distribution). This theorem was proved independently by Polish-American mathematician Mark Kac and Soviet mathematician Sergei Bernstein.

Formulation

Let ${\displaystyle \xi }$  and ${\displaystyle \eta }$  are independent random variables. If ${\displaystyle \xi +\eta }$  and ${\displaystyle \xi -\eta }$  are independent then ${\displaystyle \xi }$  and ${\displaystyle \eta }$  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.

References

• Kac M. "On a characterization of the normal distribution," American Journal of Mathematics. 1939. 61. pp. 726—728.
• Bernstein S. N. "On a property which characterizes a Gaussian distribution," Proceedings of the Leningrad Polytechnic Institute. 1941. V. 217, No 3. pp. 21—22.