Bernoulli scheme

The full n-shift corresponds to the Bernoulli scheme without the measure.

These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case.

A Bernoulli scheme with only two possible states is known as a Bernoulli process.

The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes.

It states that if two different Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic.

Most of the properties of the Bernoulli scheme follow from the countable direct product, rather than from the finite base space.

The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic.

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.

An example of a discrete-time stationary process where the sample space is also discrete (so that the random variable may take one of N possible values) is a Bernoulli scheme.

The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different measure-preserving dynamical systems can be judged to be isomorphic to Bernoulli schemes.

Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme.