What is Ergodicity?

Dated Nov 23, 2016; last modified on Mon, 05 Sep 2022

A random process is ergodic if all of its statistics can be determined from a sample function of the process. That is, the ensemble averages equal the corresponding time averages with probability one.

Role of Ergodicity in Human Inference

  • A newspaper has previously printed some inaccurate information, therefore, the newspaper is going to publish inaccurate information in the future.
    • Fair. Ensemble of published articles is more or less ergodic.
  • More crimes are committed by black persons than by white persons, therefore each individual black person is not to be trusted?
    • The ensemble of black people is not at all ergodic!

Ergodic ≠ Stationary

  • Although all ergodic processes are stationary (unconditional joint PDF doesn’t change in time; consequently parameters such as μ, σ don’t change over time), they are not equivalent.
    • Say we have coin A with \( \mathbb{P}\{H\} =.5 \) and coin B with \( \mathbb{P}\{H\} = .25 \)
    • We pick either of the coins with \( p = .5\)
    • We toss the coin that we picked repeatedly, noting \(H_1,H_2,H_n\)
    • Although the process is stationary, the time average \(H_1 + … + H_n \) either converges to \(\frac{1}{2}\) or \(\frac{1}{4} \) with equal probability.
    • The time average depends on the coin you choose, while the probabilistic average \(\left(\frac{3}{8} \right) \) is calculated for the whole system.

An Ergodic System (In a Sense)

  • Say I gave \(n\) people a die each, had them roll their die once. \( \frac{X_1 + … + X_n}{n} \) is a finite-sample average which approaches the ensemble average as \(n \to \infty \).

  • Say I rolled a die \(t\) times and calculate \(\frac{X_1 + … + X_t}{t}\). This finite-time average approaches the time average as \(t \to \infty \).

  • One implication of ergodicity is that ensemble averages will be the same as time averages.

    • In the first case, as \(n \to \infty \) the randomness is removed from the system.
    • In the second case, as \(t \to \infty \) the randomness is removed.
    • But both methods give the same answer, within errors.

Notes on Ergodicity

  • In this sense, rolling dice is an ergodic system. But if we bet on the results of rolling a die, wealth does not follow an ergodic process under typical betting rules.

    • If I go bankrupt, I’ll stay bankrupt. So the time average of my wealth will approach zero as time passes, even though the ensemble average of my wealth may increase.
  • Stationarity is required for ergodicity, for there can be no growth in an ergodic system. Ergodic systems are zero-sum games. No branching occurs in an ergodic system, no decision has any consequences because sooner or later we’ll end up in the same situation again and can reconsider.

  • The key is that most systems of interest to us, including finance, are non-ergodic.

References

  1. What is Ergodicity? Lars P. Syll. larspsyll.wordpress.com . Nov 23, 2016.