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

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Let    be independent random variables. Let    be nonzero constants such that   for all  . If the conditional distribution of the linear form   given   is symmetric then all random variables   have normal distributions (Gaussian distributions).

References

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