ergodic sequence

Let $A\; =\; \backslash \{a\_j\backslash \}$ 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 $1\backslash leq\; k\; \backslash leq\; q$, one has

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

where

:$N(A,t)\; =\; \backslash mbox\{card\}\; \backslash \{a\_j\; \backslash in\; A\; :\; a\_j\; \backslash leq\; t\; \backslash \}$

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

:$N(A,t,k,q)\; =\; \backslash mbox\{card\}\; \backslash \{a\_j\; \backslash in\; A\; :\; a\_j\backslash leq\; t,\backslash ,\; a\_j\; \backslash mod\; q\; =\; k\; \backslash \}$

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

:$\backslash lim\_\{t\backslash to\backslash infty\}\; \backslash frac\{1\}\{N(A,t)\}\; \backslash sum\_\{j;\; a\_j\backslash leq\; t\}$

\exp \frac{2\pi ika_j}{q} = 0

vanish for every integer k with $k\; \backslash mod\; q\; \backslash ne\; 0$.

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.{{cite web|url=https://www.impan.pl/~gutman/The%20Theory%20of%20Bernoulli%20Shifts.pdf|title=The theory of Bernoulli shifts|first=Paul C.|last=Shields |date=March 10, 2003|access-date=February 7, 2025|format=pdf}}

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

:$\backslash mathbb\{Z\}^\backslash omega\; =\; \backslash \{n\backslash in\; \backslash mathbb\{Z\}\; :\; X\_n(\backslash omega)\; =\; 1\; \backslash \}$

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

{{see also|Counterexample}}

Fibonacci numbers are not an ergodic sequence.{{cite journal |url=https://www.sciencedirect.com/science/article/pii/S0097316512000556|journal=Journal of Combinatorial Theory, Series A|volume=119|issue=7|date=October 1, 2012|pages=1398-1413|title=From Fibonacci numbers to central limit type theorems|first1=Steven J. |last1=Miller |first2=Yinghui |last2=Wang|access-date=February 7, 2025}}

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

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Category:Ergodic theory

Category:Integer sequences

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