Paul Lévy continuity theorem

Let $F_1,F_2,\mathrm{\dots }$ be distribution functions with characteristic functions ${\phi }_1,{\phi }_2,\mathrm{\dots }$, respectively. If ${\phi }_n$ converges pointwise to a limit $\phi$, and if $\phi (t)$ is continuous at $t=0$, then there exists a distribution function $F$ such that $F_n\to F$ weakly (http://planetmath.org/ConvergenceInDistribution), and the characteristic function associated to $F$ is $\phi$.

Remark. The reciprocal of this theorem is a corollary to the Helly-Bray theorem; hence $F_n\to F$ weakly if and only if ${\phi }_n\to \phi$ 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
Canonical name	PaulLevyContinuityTheorem
Date of creation	2013-03-22 13:14:31
Last modified on	2013-03-22 13:14:31
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	7
Author	Koro (127)
Entry type	Theorem
Classification	msc 60E10