Let $F_1,F_2,\dots$ be distribution functions with characteristic functions $\varphi_1,\varphi_2,\dots$ , respectively. If $\varphi_n$ converges pointwise to a limit $\varphi$ , and if $\varphi(t)$ is continuous at $t=0$ , then there exists a distribution function $F$ such that $F_n\rightarrow F$ weakly, and the characteristic function associated to $F$ is $\varphi$ .

Remark. The reciprocal of this theorem is a simple corollary to the Helly-Bray theorem; hence $F_n\rightarrow F$ weakly if and only if $\varphi_n\rightarrow\varphi$ 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.