# Kolmogorov’s inequality

Let ${X}_{1},\mathrm{\dots},{X}_{n}$ be independent^{} random variables^{} in a probability space^{}, such that $\mathrm{E}[{X}_{k}]=0$ and $$ for
$k=1,\mathrm{\dots},n$. Then, for each $\lambda >0$,

$$P(\underset{1\le k\le n}{\mathrm{max}}|{S}_{k}|\ge \lambda )\le \frac{1}{{\lambda}^{2}}\mathrm{Var}[{S}_{n}]=\frac{1}{{\lambda}^{2}}\sum _{k=1}^{n}\mathrm{Var}[{X}_{k}],$$ |

where ${S}_{k}={X}_{1}+\mathrm{\cdots}+{X}_{k}$.

Title | Kolmogorov’s inequality |
---|---|

Canonical name | KolmogorovsInequality |

Date of creation | 2013-03-22 13:13:15 |

Last modified on | 2013-03-22 13:13:15 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 9 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 60E15 |

Related topic | ChebyshevsInequality2 |

Related topic | MarkovsInequality |

Related topic | ChebyshevsInequality |