What is a good example of an ergodic process?

I'm trying to find simple examples of an ergodic process. What process comes to your mind as a good illustration of its properties?

A quick research (Wikipedia, another answer) mainly gives examples of non-ergodic processes. Also, I'm wondering which real world phenomena lend itself to be modeled as an ergodic process?

Answer

Just suppose I give you a series of numbers, and I tell you they were picked randomly. And you know I am not trying to deceive you. Numbers are: $3$, $1$, $4$, $1$, $5$, $3$, $2$, $3$, $4$, $3$.

I now propose you to predict the next one, or at least, to be as close as possible. Which number would you pick?

[Think]

[Compute]

• I bet most of the readers are likely to choose a number between $0$ and $6$. Because of the limited span.


• Perhaps an integer. Who is likely to propose $\pi$ (even thinking of the first digits)?



• Possibly $2$, $3$, or $4$. Maybe even $3$.

Basically, you are assuming that I provided numbers with some unknown rule. And perhaps, you may think (or make the hypothesis) that the series of given numbers, if long enough, can provide you with a good understanding of the rules that I have in mind. If you do so, you hypothesize that my mental process is ergodic:

a process in which every sequence or sizable sample is equally representative of the whole (as in regard to a statistical parameter) (Merriam-Webster)

Here, there is no way to be sure that my series follows an ergodic process. 3432 is my card PIN, 3 a mistake (I intended 6, but I am clumsy), 4, 3, 1 and 5 are first digits of $\pi$ that I use quite often. My next "number" would have been C (in hexadecimal). I do not believe this process is ergodic. Each number comes out from different laws. But honestly, I do not know. Maybe I am subject to some higher order forces that drive me under ergodicity rules.

So, ergodicity is a hypothesis of a sort of "simplicity" in the rules of a process. Like stationarity or sparsity. Cast a regular die with $6$ faces. Toss a normal coin. If nothing outside tries to influence the result (an invisible being that catches the die and shows some face of its choice), you are likely to produce an ergodic process.

Instead of being able to toss an infinite number of coins, with your infinite number of thumbs, precisely at the same second, you toss one coin every second, and believe the final result is about the same.

The Brownian motion possesses ergodic properties too.