Kac–Bernstein theorem

Let ${\displaystyle \xi }$ and ${\displaystyle \eta }$ be independent random variables. If ${\displaystyle \xi +\eta }$ and ${\displaystyle \xi -\eta }$ are independent, then ${\displaystyle \xi }$ and ${\displaystyle \eta }$ are normally distributed (Gaussian).

A generalization of the Kac–Bernstein theorem is the Darmois–Skitovich theorem, in which the normal distribution is characterized by the independence of two linear forms of ${\displaystyle n}$ independent random variables.

The first works devoted to generalizing the Kac–Bernstein theorem to locally compact Abelian groups are due to A. L. Rukhin^[3] and also to H. Heyer and Ch. Rall^[4]. These works studied the following problem. Let ${\displaystyle X}$ be a locally compact Abelian group, and let ${\displaystyle \xi }$ and ${\displaystyle \eta }$ be independent random variables taking values in ${\displaystyle X}$ with distributions ${\displaystyle \mu }$ and ${\displaystyle \nu }$. For which groups ${\displaystyle X}$ does the independence of ${\displaystyle \xi +\eta }$ and ${\displaystyle \xi -\eta }$ imply that ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are convolutions of Gaussian and idempotent distributions? Some sufficient conditions were established in ^[3] and ^[4]. The final result was obtained by G. M. Feldman, who proved the following theorem.

Theorem^[5]; see also ^[6]. Let ${\displaystyle X}$ be a second countable locally compact Abelian group. Let ${\displaystyle \xi }$ and ${\displaystyle \eta }$ be independent random variables taking values in ${\displaystyle X}$ with distributions ${\displaystyle \mu }$ and ${\displaystyle \nu }$. Then the independence of ${\displaystyle \xi +\eta }$ and ${\displaystyle \xi -\eta }$ implies that ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are convolutions of Gaussian and idempotent distributions if and only if the connected component of the zero of the group ${\displaystyle X}$ contains no elements of order 2.