In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an ergodic sequence is a certain type of integer sequence

Integer sequence

In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...

, having certain equidistribution properties.

Definition

Let be an infinite, strictly increasing sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 of positive integers. Then, given an integer q, this sequence is said to be ergodic mod q if, for all integers , one has


where


and card is the count (the number of elements) of a set, so that is the number of elements in the sequence A that are less than or equal to t, and


so is the number of elements in the sequence A, less than t, that are equivalent to k modulo q. That is, a sequence is an ergodic sequence if it becomes uniformly distributed mod q as the sequence is taken to infinity.

An equivalent definition is that the sum


vanish for every integer k with .

If a sequence is ergodic for all q, then it is sometimes said to be ergodic for periodic systems.

Examples

The sequence of positive integers is ergodic for all q.

Almost all

Almost all

In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

 Bernoulli sequences, that is, sequences associated with a Bernoulli process

Bernoulli process

In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identical and independent...

, are ergodic for all q. That is, let be a probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

 of random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s over two letters . Then, given , the random variable is 1 with some probability p and is zero with some probability 1-p; this is the definition of a Bernoulli process. Associated with each is the sequence of integers


Then almost every sequence is ergodic.

See also

• Ergodic theory
  Ergodic theory
  Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
  
  
• Ergodic process
  Ergodic process
  In signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process.- Specific definitions :...
  
  , for the use of the term in signal processing
  Signal processing
  Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...