# 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 |