Heyde theorem

In the mathematical theory of probability, the Heyde theorem is the characterization theorem concerning the normal distribution (the Gaussian distribution) by the symmetry of one linear form given another. This theorem was proved by C. C. Heyde.

Formulation

Let $\xi _{j},j=1,2,\ldots ,n,n\geq 2$ be independent random variables. Let $\alpha _{j},\beta _{j}$ be nonzero constants such that ${\frac {\beta _{i}}{\alpha _{i}}}+{\frac {\beta _{j}}{\alpha _{j}}}\neq 0$ for all $i\neq j$ . If the conditional distribution of the linear form $L_{2}=\beta _{1}\xi _{1}+\cdots +\beta _{n}\xi _{n}$ given $L_{1}=\alpha _{1}\xi _{1}+\cdots +\alpha _{n}\xi _{n}$ is symmetric then all random variables $\xi _{j}$ have normal distributions (Gaussian distributions).