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 , 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 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 is the cylinder sets. Given a cylinder set , its measure is The equivalent expression, using the notation of probability theory, is for the random variables 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.

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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 because it states that many systems previously believed to be unrelated are in fact isomorphic; these include all finite stationary stochastic processes, including Markov chains and subshifts of finite type, Anosov flows and Sinai's billiards, ergodic automorphisms of the n-torus, and the continued fraction transform.

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