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 occuring with probability p_i, with , and Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In probability and statistics, a Bernoulli process is a discrete_time stochastic process consisting of finite or infinite sequence of independent random variables X1, X2, X3,..., such that For each i, the value of Xi is either 0 or 1; For all values of i, the probability that Xi = 1 is... Discrete time is non-continuous time. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

.

The sample space is usually denoted as In probability theory, the sample space, often denoted S, Ω or U (for universe), of an experiment or random trial is the set of all possible outcomes. ...

as a short-hand for

The associated measure is In mathematics, a measure is a function that assigns a number, e. ...

The σ-algebra 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 In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...

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 In mathematics, a measure is a function that assigns a number, e. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...

Tx_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 In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ...

is a measure-preserving dynamical system, and is called the Bernoulli scheme. It is often denoted by In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ...

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

The N=2 Bernoulli scheme is called a Bernoulli process. In probability and statistics, a Bernoulli process is a discrete_time stochastic process consisting of finite or infinite sequence of independent random variables X1, X2, X3,..., such that For each i, the value of Xi is either 0 or 1; For all values of i, the probability that Xi = 1 is...

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. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... This article may be too technical for most readers to understand. ... P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ...

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