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 In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (e ...
, and are thus important in the study of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s. Many important dynamical systems (such as
Axiom A systems) exhibit a
repellor that is the product of the
Cantor set and a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the
Markov partition
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete ...
. The term ''shift'' is in reference to the
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift ...
, which may be used to study Bernoulli schemes. The
Ornstein isomorphism theorem In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important b ...
shows that Bernoulli shifts are isomorphic when their
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
is equal.
Definition
A Bernoulli scheme is a
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
stochastic process where each
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
random variable may take on one of ''N'' distinct possible values, with the outcome ''i'' occurring with probability
, with ''i'' = 1, ..., ''N'', and
:
The
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
is usually denoted as
:
as a shorthand for
:
The associated
measure is called the Bernoulli measure
:
The
σ-algebra on ''X'' is the product sigma algebra; that is, it is the (countable)
direct product of the σ-algebras of the finite set . Thus, the triplet
:
is a
measure space. A basis of
is the
cylinder sets. Given a cylinder set