Let be distribution functions with characteristic functions , respectively. If converges pointwise to a limit , and if is continuous at , then there exists a distribution function such that weakly, and the characteristic function associated to is .

Remark. The reciprocal of this theorem is a simple corollary to the Helly-Bray theorem; hence weakly if and only if pointwise; but this theorem says something stronger than the sufficiency of that proposition: it says that the limit of a sequence of characteristic functions is a characteristic function whenever it is continuous at 0.