Bernoulli Scheme - Definition

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, with i = 1, ..., N, and

The sample space is usually denoted as

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 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. The elements of are commonly called cylinder sets. Given a cylinder set, its measure is

\mu\left(\right)= \prod_{i=0}^n p_{x_i}

The equivalent expression, using the notation of probability theory, is

\mu\left(\right)= \mathrm{Pr}(X_0=x_0, X_1=x_1, \ldots, X_n=x_n)

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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