Bernoulli scheme

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In mathematics, the Bernoulli scheme is a generalization of the Bernoulli process to more than two possible outcomes. That is, it 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 $p i$ , with $i=1,\ldots,N$ , and

$\sum_{i=1}^N p_i = 1$ .

The sample space is usually denoted as

$X=\{1,\ldots,N \}^\mathbb{Z}$

as a short-hand for

$X=\{ x=(\ldots,x_{-1},x_0,x_1,\ldots) : x_k \in \{1,\ldots,N\} \; \forall k \in \mathbb{Z} \}$

The associated measure is

$\mu = \{p_1,\ldots,p_N\}^\mathbb{Z}$

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

$(X,\mathcal{A},\mu)$

is a measure space. The Bernoulli scheme, as any stochastic process, may be viewed as a dynamical system by endowing it with the shift operator T where

T x k = x k + 1

Since the probabilities $p i$ of each outcome are independent, the shift preserves the measure, and thus T is a measure-preserving transformation. The quadruplet

$(X,\mathcal{A},\mu, T)$

is a measure-preserving dynamical system, and is called the Bernoulli scheme. It is often denoted by

$BS(p)=BS(p_1,\ldots,p_N)$

The N=2 Bernoulli scheme is called a Bernoulli process.

[edit] Properties

The Bernoulli scheme is a stationary stochastic process, and, conversely, every stationary stochastic process is a Bernoulli scheme.

Ya. Sinai demonstrated that the Kolmogorov entropy of a Bernoulli scheme is given by

$h = -\sum_{i=1}^N p_i \log p_i$

The isomorphism theorem for Bernoulli schemes, sometimes called the Ornstein isomorphism theorem, proven by D. S. Ornstein in 1968, states that two Bernoulli schemes with the same entropy are isomorphic. By isomorphic, it is meant that if X and Y are two sample spaces, then there exists a function between these two that is measurable and invertible, that commutes with the measures, and that commutes with the shift operators for almost all sequences in X and Y. A simplified proof of the isomorphism theorem was given by Michael S. Keane and M. Smorodinsky in 1979.

When N is a prime number, sequences in the sample space may be represented by p-adic numbers. If the probabilities are uniform, that is, each $p i = 1 / N$ , then the distribution of sequences corresponds to a uniform measure on the space of numbers. As a result, the results from p-adic analysis may be applied.

[edit] See also

[edit] References

Michael S. Keane, Ergodic theory and subshifts of finite type, (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 0-19-853390-X
D.S. Ornstein, "Ornstein isomorphism theorem" SpringerLink Encyclopaedia of Mathematics (2001)

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Categories: Ergodic theory | Stochastic processes