# strong law of large numbers

A sequence of random variables^{} ${X}_{1},{X}_{2},\mathrm{\dots}$ with finite expectations
in a probability space^{} is said to satisfiy the strong law of large numbers^{} if

$$\frac{1}{n}\sum _{k=1}^{n}({X}_{k}-\mathrm{E}[{X}_{k}])\stackrel{a.s.}{\to}0,$$ |

where $a.s.$ stands for convergence almost surely.

When the random variables are identically distributed, with expectation $\mu $, the law becomes:

$$\frac{1}{n}\sum _{k=1}^{n}{X}_{k}\stackrel{a.s.}{\to}\mu .$$ |

Kolmogorov’s strong law of large numbers theorems give conditions on the random variables under which the law is satisfied.

Title | strong law of large numbers |
---|---|

Canonical name | StrongLawOfLargeNumbers |

Date of creation | 2013-03-22 13:13:10 |

Last modified on | 2013-03-22 13:13:10 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 11 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 60F15 |

Related topic | MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables |