Strong mixing

In mathematics, strong mixing is a concept applied in ergodic theory, i.e. the study of dynamical systems at the level of measure theory. It can be applied to stochastic processes.

Let

$\langle X_t \rangle = \{ \ldots, X_{t-1}, X_t, X_{t+1}, \ldots \}$

be a sequence of random variables, and

$X_a^b$

the sigma-algebra generated by

$\{X_a, X_{a+1},\ldots, X_b \}$

for

$-\infty \leq a \leq b \leq \infty$ .

$\langle X_t \rangle$ is strong mixing if

$\alpha(s)\downarrow 0$

$s\rightarrow \infty$ ,

where

$\alpha(s)\equiv \sup_{-\infty < t < \infty, A\in X_{-\infty}^t, B\in X_{t+s}^\infty } |P(A \cap B) - P(A)P(B)|$

is the so-called strong mixing coefficient.