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.
Definition.
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 formula_1, 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 formula_6 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 formula_6 is the cylinder sets. Given a cylinder set formula_9, its measure is
The equivalent expression, using the notation of probability theory, is
for the random variables formula_12
The Bernoulli scheme, as any stochastic process, may be viewed as a dynamical system by endowing it with the shift operator "T" where
Since the outcomes are independent, the shift preserves the measure, and thus "T" is a measure-preserving transformation. The quadruplet
is a measure-preserving dynamical system, and is called a Bernoulli scheme or a Bernoulli shift. It is often denoted by
The "N" = 2 Bernoulli scheme is called a Bernoulli process. The Bernoulli shift can be understood as a special case of the Markov shift, where all entries in the adjacency matrix are one, the corresponding graph thus being a clique.
Matches and metrics.
The Hamming distance provides a natural metric on a Bernoulli scheme. Another important metric is the so-called formula_16 metric, defined via a supremum over "string matches".
Let formula_17 and formula_18 be two strings of symbols. A match is a sequence "M" of pairs formula_19 of indexes into the string, i.e. pairs such that formula_20 understood to be totally ordered. That is, each individual subsequence formula_21 and formula_22 are ordered: formula_23 and likewise formula_24
The formula_16-"distance" between formula_26 and formula_27 is
where the supremum is being taken over all matches formula_29 between formula_26 and formula_27. This satisfies the triangle inequality only when formula_32 and so is not quite a true metric; despite this, it is commonly called a "distance" in the literature.
Generalizations.
Most of the properties of the Bernoulli scheme follow from the countable direct product, rather than from the finite base space. Thus, one may take the base space to be any standard probability space formula_33, and define the Bernoulli scheme as
This works because the countable direct product of a standard probability space is again a standard probability space.
As a further generalization, one may replace the integers formula_35 by a countable discrete group formula_36, so that 
For this last case, the shift operator is replaced by the group action
for group elements formula_39 and formula_40 understood as a function formula_41 (any direct product formula_42 can be understood to be the set of functions formula_43, as this is the exponential object). The measure formula_44 is taken as the Haar measure, which is invariant under the group action: 
These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case.
Properties.
Ya. Sinai demonstrated that the Kolmogorov entropy of a Bernoulli scheme is given by
This may be seen as resulting from the general definition of the entropy of a Cartesian product of probability spaces, which follows from the asymptotic equipartition property. For the case of a general base space formula_47 ("i.e." a base space which is not countable), one typically considers the relative entropy. So, for example, if one has a countable partition formula_48 of the base "Y", such that formula_49, one may define the entropy as
In general, this entropy will depend on the partition; however, for many dynamical systems, it is the case that the symbolic dynamics is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition.
Ornstein Isomorphism Theorem.
The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic. The result is sharp, in that very similar, non-scheme systems, such as Kolmogorov automorphisms, do not have this property.
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. The result was surprising, as many systems previously believed to be unrelated proved to be isomorphic. These include all finite stationary stochastic processes, subshifts of finite type, finite Markov chains, Anosov flows, and Sinai's billiards: these are all isomorphic to Bernoulli schemes.
For the generalized case, the Ornstein isomorphism theorem still holds if the group "G" is a countably infinite amenable group.
Bernoulli automorphism.
An invertible, measure-preserving transformation of a standard probability space (Lebesgue space) is called a Bernoulli automorphism if it isomorphic to a Bernoulli shift.
Loosely Bernoulli.
A system is termed "loosely Bernoulli" if it is Kakutani-equivalent to a Bernoulli shift; in the case of zero entropy, if it is Kakutani-equivalent to an irrational rotation of a circle.