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.

Read more about this topic:  Bernoulli Scheme

Famous quotes containing the word definition:

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)