Ergodic sequence

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

Definition[edit]

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.

Examples[edit]

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.

See also[edit]