mixing

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

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

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 $A,B\in 𝒜$.

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