Bernoulli scheme

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal.

A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of N distinct possible values, with the outcome i occurring with probability ${\displaystyle p_{i}}$, with i = 1, ..., N, and

The sample space is usually denoted as

as a shorthand for

The associated measure is called the Bernoulli measure

The σ-algebra ${\displaystyle {\mathcal {A}}}$ on X is the product sigma algebra; that is, it is the (countable) direct product of the σ-algebras of the finite set {1, ..., N}. Thus, the triplet

is a measure space. A basis of ${\displaystyle {\mathcal {A}}}$ is the cylinder sets. Given a cylinder set ${\displaystyle [x_{0},x_{1},\ldots ,x_{n}]}$, its measure is

The equivalent expression, using the notation of probability theory, is