Paul Lévy continuity theorem

Let F1,F2, be distribution functionsMathworldPlanetmath with characteristic functionsMathworldPlanetmathPlanetmathPlanetmathPlanetmath φ1,φ2,, respectively. If φn converges pointwise to a limit φ, and if φ(t) is continuous at t=0, then there exists a distribution function F such that FnF weakly (, and the characteristic function associated to F is φ.

Remark. The reciprocal of this theorem is a corollary to the Helly-Bray theorem; hence FnF weakly if and only if φnφ 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