Ergodic sequence

In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties.^[1]

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

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

where

${\displaystyle N(A,t)={\mbox{card}}\{a_{j}\in A:a_{j}\leq t\}}$

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

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

so ${\displaystyle N(A,t,k,q)}$ 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

${\displaystyle \lim _{t\to \infty }{\frac {1}{N(A,t)}}\sum _{j;a_{j}\leq t}\exp {\frac {2\pi ika_{j}}{q}}=0}$

vanish for every integer k with ${\displaystyle k\mod q\neq 0}$.

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

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.^[2]

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

${\displaystyle \mathbb {Z} ^{\omega }=\{n\in \mathbb {Z} :X_{n}(\omega )=1\}}$

Then almost every sequence ${\displaystyle \mathbb {Z} ^{\omega }}$ is ergodic.

Fibonacci numbers are not an ergodic sequence.^[3]

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

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