World Library  
Flag as Inappropriate
Email this Article

Ergodic process

Article Id: WHEBN0018104627
Reproduction Date:

Title: Ergodic process  
Author: World Heritage Encyclopedia
Language: English
Subject: Stationary ergodic process, Coherence (signal processing), Ergodic theory, Asymptotic theory (statistics), Autocorrelation
Collection: Ergodic Theory, Signal Processing
Publisher: World Heritage Encyclopedia

Ergodic process

In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties (such as its mean and variance) can be deduced from a single, sufficiently long sample (realization) of the process. The reasoning behind this is that any sample from the process must represent the average statistical properties of the entire process, so that regardless what sample you choose, it represents the whole process, and not just that section of the process. A process that changes erratically at an inconsistent rate is not said to be ergodic.[1]


  • Specific definitions 1
  • Discrete-time random processes 2
  • Example of a non-ergodic random process 3
  • See also 4
  • Notes 5
  • References 6

Specific definitions

One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process X(t) has constant mean

\mu_X= E[X(t)],

and autocovariance

r_X(\tau) = E[(X(t)-\mu_X) (X(t+\tau)-\mu_X)],

that depends only on the lag \tau and not on time t. The properties \mu_X and r_X(\tau) are ensemble averages not time averages.

The process X(t) is said to be mean-ergodic[2] or mean-square ergodic in the first moment[3] if the time average estimate

\hat{\mu}_X = \frac{1}{T} \int_{0}^{T} X(t) \, dt

converges in squared mean to the ensemble average \mu_X as T \rightarrow \infty.

Likewise, the process is said to be autocovariance-ergodic or mean-square ergodic in the second moment[3] if the time average estimate

\hat{r}_X(\tau) = \frac{1}{T} \int_{0}^{T} [X(t+\tau)-\mu_X] [X(t)-\mu_x] \, dt

converges in squared mean to the ensemble average r_X(\tau), as T \rightarrow \infty. A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.[3]

An important example of an ergodic processes is the stationary Gaussian process with continuous spectrum.

Discrete-time random processes

The notion of ergodicity also applies to discrete-time random processes X[n] for integer n.

A discrete-time random process X[n] is ergodic in mean if

\hat{\mu}_X = \frac{1}{N} \sum_{n=1}^{N} X[n]

converges in squared mean to the ensemble average E[X], as N \rightarrow \infty.

Example of a non-ergodic random process

Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins, and then perform a sequence of independent tosses of our selected coin. Let X[n] denote the outcome of the nth toss, with 1 for heads and 0 for tails. Then the ensemble average is ½ · ½ + ½ · 1 = ¾; yet the long-term average is ½ for the fair coin and 1 for the two-headed coin. Hence, this random process is not ergodic in mean.

See also


  1. ^ Originally due to L. Boltzmann. See part 2 of ('Ergoden' on p.89 in the 1923 reprint.) It was used to prove equipartition of energy in the kinetic theory of gases
  2. ^ Papoulis, p.428
  3. ^ a b c Porat, p.14


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.