# mixing

Let $f$ be a measure-preserving transformation^{} of a probability space $(X,\mathcal{A},\mu )$. We say that $f$ is *mixing* (or strong-mixing) if for all $A,B\in \mathcal{A}$,

$$\underset{n\to \mathrm{\infty}}{lim}\mu ({f}^{-n}(A)\cap B)=\mu (A)\mu (B),$$ |

and $f$ is *weakly mixing* if

$$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\sum _{i=0}^{n-1}|\mu ({f}^{-i}(A)\cap B)-\mu (A)\mu (B)|=0$$ |

for all $A,B\in \mathcal{A}$.

Every mixing transformation is weakly mixing, and every weakly mixing transformation is ergodic.

Title | mixing |

Canonical name | Mixing |

Date of creation | 2013-03-22 14:06:34 |

Last modified on | 2013-03-22 14:06:34 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37A25 |

Defines | strongly mixing |

Defines | strong mixing |

Defines | strong-mixing |

Defines | weak-mixing |

Defines | weakly mixing |

Defines | weak mixing |