Ergodic Sequence
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ergodic sequence is a certain type of
integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
, having certain equidistribution properties.


Definition

Let A = \ be an infinite, strictly increasing
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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_ \frac = \frac where :N(A,t) = \mbox \ and
card Card or The Card may refer to: * Various types of plastic cards: **By type ***Magnetic stripe card *** Chip card *** Digital card **By function ***Payment card ****Credit card **** Debit card ****EC-card ****Identity card ****European Health Insur ...
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 \ 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_ \frac \sum_ \exp \frac = 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

The sequence of positive integers is ergodic for all ''q''.
Almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
Bernoulli sequence In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. Th ...
s, that is, sequences associated with a
Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. Th ...
, are ergodic for all ''q''. That is, let (\Omega,Pr) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s over two letters \. 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^\omega = \ Then almost every sequence \mathbb^\omega is ergodic.


See also

*
Ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
*
Ergodic process In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must ...
, for the use of the term in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
{{DEFAULTSORT:Ergodic Sequence Ergodic theory Integer sequences