# Helly-Bray theorem

Let $F,{F}_{1},{F}_{2},\mathrm{\dots}$ be distribution functions^{}. If ${F}_{n}$ converges weakly (http://planetmath.org/ConvergenceInDistribution) to $F$, then

$${\int}_{\mathbb{R}}g(x)\mathit{d}{F}_{n}(x)\underset{n\to \mathrm{\infty}}{\overset{}{\to}}{\int}_{\mathbb{R}}g(x)\mathit{d}F(x)$$ |

for each continuous^{} bounded function $g:\mathbb{R}\to \mathbb{R}$.

*Remark.* The integrals involved are Riemann-Stieltjes integrals.

Title | Helly-Bray theorem |
---|---|

Canonical name | HellyBrayTheorem |

Date of creation | 2013-03-22 13:14:18 |

Last modified on | 2013-03-22 13:14:18 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 60E05 |