# Ergodic sequence

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

## Definition

Let $A=\{a_{j}\}$ 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 $1\leq k\leq q$ , one has

$\lim _{t\to \infty }{\frac {N(A,t,k,q)}{N(A,t)}}={\frac {1}{q}}$ where

$N(A,t)={\mbox{card}}\{a_{j}\in A:a_{j}\leq t\}$ and card is the count (the number of elements) of a set, so that $N(A,t)$ is the number of elements in the sequence A that are less than or equal to t, and

$N(A,t,k,q)={\mbox{card}}\{a_{j}\in A:a_{j}\leq t,\,a_{j}\mod q=k\}$ so $N(A,t,k,q)$ 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

$\lim _{t\to \infty }{\frac {1}{N(A,t)}}\sum _{j;a_{j}\leq t}\exp {\frac {2\pi ika_{j}}{q}}=0$ vanish for every integer k with $k\mod q\neq 0$ .

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; that is, let $(\Omega ,Pr)$ be a probability space of random variables over two letters $\{0,1\}$ . Then, given $\omega \in \Omega$ , the random variable $X_{j}(\omega )$ 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 $\omega$ is the sequence of integers

$\mathbb {Z} ^{\omega }=\{n\in \mathbb {Z} :X_{n}(\omega )=1\}$ Then almost every sequence $\mathbb {Z} ^{\omega }$ is ergodic.