Ergodic sequence

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

Definition

Let $$A = {a_j} be an infinite, strictly increasing sequence of positive integers. Then, given an integer "q", this sequence is said to be ergodic mod "q" if, for all integers $$1leq k leq q, one has

:$$lim_{t oinfty} 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_jleq 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 oinfty} frac{1}{N(A,t)} sum_{j; a_jleq t} exp frac{2pi ika_j}{q} = 0

vanish for every integer "k" with $$k mod q e 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 = {nin mathbb{Z} : X_n(omega) = 1 }

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

ee also

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