In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties.

Definition

be an infinite, strictly increasing sequence of positive integers. Then, given an integer q, this sequence is said to be ergodic mod q , one has

where

and card is the number of elements in the sequence A that are less than or equal to t, and

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.

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 Bernoulli sequences, that is, sequences associated with a Bernoulli process, are ergodic for all q.

be a probability space of random variables is 1 with some probability p and is zero with some probability 1-p is the sequence of integers

is ergodic.

Not ergodic

Fibonacci numbers are not an ergodic sequence.

See also

• Ergodic theory

• Ergodic process, for the use of the term in signal processing

References