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I've been told that Ergodicity gives us a practical vision of processes WSS (Wise-sense stationary) and a bunch of integrals. For me, it is not enough to fully understand it.

Could someone explain me Ergodicity in a simple way?

EDIT:

Thank you all for those interested in the question and answered, here i will share an example:

y(t) is a random process where {i(t),q(t)} are two random stationary processes, incorrelated, null mean and autocorrelation Ri(z) = Rq(z).

y(t) = i(t)cos(2πf0t)−q(t)sin(2πf0t)

The exercise asks for mean and autocorrelation of y(t) and finally if that process is stationary or cyclostationary.

I have resolved that already but, what about ergodicity.

Is this process ergodic? How could i demonstrate such thing?

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  • $\begingroup$ It's not clear what you don't understand, please clarify the question. What is the purpose and what kind of research have you done? $\endgroup$ – cherub May 7 '18 at 16:44
  • $\begingroup$ As pointed out by @cherub, "a bunch of integrals" is not a very accurate way of describing the things you don't understand. $\endgroup$ – Emil May 8 '18 at 11:09
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Here's the simplest way I can think of: if you watch a stochastic process long enough you're going to see every possible outcome. Not only that, but also you can obtain the probabilities of such outcomes.

What's the deal here? There are some processes where you can't have repeated trials. For instance, a coin toss is easy to replicate cross-sectionally: just get many coins, and toss them simultaneously. What about weather? Can you replicate weather on Jan 1 2018? Obviously, no. There's only one Jan 1 2018, and it will never repeat. However, if you had ergodicity you could watch weather for many days or even years, and figure what were the probabilities of different weather realization on Jan 1 2018.

Summarizing, ergodicity establishes certain equivalence between multiple trials in the same time period (cross-sectional) and prolonged observation of the same process over time (time-series). This is helpful when, particularly, cross-sectional experiment is not possible and observation over time is possible.

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  • $\begingroup$ This really helped me :D $\endgroup$ – WhiteGlove Dec 30 '18 at 16:56
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To quote Wiki with a little formatting of my own:

[...] a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process.

In other, perhaps overly simplified words, if you observe a process for a long enough time, then you can know all there is to know about the process in terms of statistical behavior, because what you are observing gets closer and closer (coverges) to the "true" (what the wiki calls ensemble) properties of the process properties.

On the same page, the counterexample with the two coins where one is fair and the other has only one outcome shows you exactly a process that is not mean-ergodic: when you pick a coin at random and start throwing it, the average of those throws will never be close to the (ensemble) mean of the process, no matter which coin you ended up picking or how many times you throw it.

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From Bishop, a Markov chain is ergodic when you can run it starting from any initial distribution and end up converging to its invariant distribution (steady state, or equilibrium).

A sufficient condition for ergodicity is that you can move from any state to any other with nonzero probability in a finite number of steps.

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    $\begingroup$ I wonder if it might be necessary to add "in finite time" after the nonzero probability part. Even if the chain is made of recurrent states, these states still need to be positive recurrent, no? $\endgroup$ – Emil May 7 '18 at 17:07

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