Editing Mixing (mathematics) (section)

=== Generalizations ===

The definition given above is sometimes called '''strong 2-mixing''', to distinguish it from higher orders of mixing.  A '''strong 3-mixing system''' may be defined as a system for which
: <math>\lim_{m,n\to\infty} \mu (A \cap T^{-m}B \cap T^{-m-n}C) = \mu(A)\mu(B)\mu(C)</math>
holds for all measurable sets ''A'', ''B'', ''C''.  We can define '''strong k-mixing''' similarly.  A system which is '''strong''' ''k''-'''mixing''' for all ''k''&nbsp;=&nbsp;2,3,4,... is called '''mixing of all orders'''.

It is unknown whether strong 2-mixing implies strong 3-mixing.  It is known that strong ''m''-mixing implies [[ergodicity]].