Bernoulli scheme

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.^[1][2] Bernoulli schemes are important in the study of dynamical systems, as most such 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.^[3] 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^[4] shows that Bernoulli shifts are isomorphic when their entropy is equal. Finite stationary stochastic processes are isomorphic to the Bernoulli shift; in this sense, Bernoulli shifts are universal.

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 p_i, with i = 1, ..., 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

Tx_k = x_{k + 1}.

Since the outcomes 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 a Bernoulli scheme or a Bernoulli shift. It is often denoted by

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

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.

Properties

The Bernoulli scheme is a stationary stochastic process; conversely, all finite^{[clarification needed]} stationary stochastic processes, including subshifts of finite type and finite Markov chains, are Bernoulli schemes; this is essentially the content of the Ornstein isomorphism theorem.^{[citation needed]}

Ya. Sinai^{[citation needed]} 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 Donald Ornstein in 1968,^{[citation needed]} states that two Bernoulli schemes with the same entropy are isomorphic. Here isomorphic means 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,^{[clarification needed]} and that commutes with the shift operators^{[clarification needed]} 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.^{[citation needed]}

When N is a prime number, sequences in the sample space may be represented by p-adic numbers.^{[citation needed]} 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.^{[clarification needed]} As a result, the results from p-adic analysis may be applied.^{[clarification needed]}

See also

• Shift of finite type
• Markov chain
• Hidden Bernoulli model

References

1. ^ P. Shields, The theory of Bernoulli shifts , Univ. Chicago Press (1973)
2. ^ 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
3. ^ Pierre Gaspard, Chaos, scattering and statistical mechanics(1998), Cambridge University press
4. ^ D.S. Ornstein (2001), "Ornstein isomorphism theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/O/o120070.htm