Solved – Ergodicity explained in layman terms

I've been told that Ergodicity gives us a practical vision of processes WSS (Wide-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?


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 $Ri(z) = Rq(z)$.

$$y(t) = i(t)cos(2pi f_0t)−q(t)sin(2pi f_0t)$$

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?

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.

If you live long enough, you'll experience everything.

Robert Torricelli,

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