Ergodic theory#Ergodic theorems

The ergodic theorem is an important set of stochastics. It provides a form of the law of large numbers for dependent random variables and provides the mathematical basis of the ergodic hypothesis of statistical physics.

Formulation of the ergodic theorem of Birkhoff

Is an integrable random variable ( ie it has a finite expectation value) and a maßerhaltende transformation on the underlying probability space (ie for all ). Then the agents converge

For almost surely to a random variable Y.

Y can be measured with respect to that of the T- invariant sets A (ie, ( A) = A) generated σ - algebra are elected and can be represented as conditional expectations.

If T is ergodic, then almost surely constant equal to the expected value of X.

The example of a stationary process

The random variables ( ) form a stationary stochastic process, ie is distributed as. Conversely, any stationary stochastic process representing in this way, if one assumes is that and from the mold. ( If this is not the case, can the picture space with the size of look instead of and. ) Where, and the " displacement mapping ", under which is mapped to, is the maßerhaltende transformation.

If have a finite expectation value converges after the ergodic theorem thus

For almost surely to a random variable. is the conditional expectation of each. If ergodicity is present, is almost surely constant, ie