# Paul Lévy continuity theorem

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 (http://planetmath.org/ConvergenceInDistribution), and the characteristic function associated to $F$ is $\varphi$.

Remark. The reciprocal of this theorem is a 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 : it says that the limit of a sequence of characteristic functions is a characteristic function whenever it is continuous at 0.

Title Paul Lévy continuity theorem PaulLevyContinuityTheorem 2013-03-22 13:14:31 2013-03-22 13:14:31 Koro (127) Koro (127) 7 Koro (127) Theorem msc 60E10