Ergodic sequence
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In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties.
[edit] Definition
Let A = {aj} 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 N(A,t) is the number of elements in the sequence A that are less than or equal to t, and
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
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.
[edit] 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 (Ω,Pr) be a probability space of random variables over two letters {0,1}. Then, given 
, the random variable Xj(ω) 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 ω is the sequence of integers
Then almost every sequence 
is ergodic.
