Ergodic sequence

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

Definition[edit]

Let be an infinite, strictly increasing 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[edit]

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. That is, let be a probability space of random variables 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[edit]