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 $A = {a j}$ 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 $k \mod q \ne 0$ .

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 $(Ω, P r)$ be a probability space of random variables over two letters ${0,1}$ . Then, given $\omega \in \Omega$ , the random variable $X j (ω)$ 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