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Is it possible to define the ergodicity of the random signal in an intuitive sense without using any statistical reference?

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2 Answers 2

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When a process or system in which the average, over time, of the physical quantities that describe it coincides with the average calculated over a set of possible states of the process or of the system itself.

I would use an example like the following:

-You would have the similar results after flipping 100 coins as flipping a coin 100 times.

I hope it helps.

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    $\begingroup$ Thanks a million. That example really helps. $\endgroup$ Jan 4, 2021 at 2:59
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    $\begingroup$ Be a bit careful, though, the definition "ensemble and time average of the mean are the same" is a very narrow definition of ergodicity; I'd call that "ergodic (only) w.r.t. averaging"(it also has the special name of "pointwise ergodicity",iirc); the definition from literature that I find most useful is that "A stochastic process (random signal) $X(t)$ is ergodic with respect to a function $g(\cdot)$ iff the time average $\overline{g(x(t)}$ of every realization converges to the ensemble average $E(g(X(t)))$". For that to exist, the latter must be a constant, so stationarity is a requirement. $\endgroup$ Jan 4, 2021 at 8:11
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The intuition about ergodicity is that past events do not alter the "probabilistic model". Hence, one can start studying the process at any moment irrespective of the past.

For example, assume $b_1, b_2,\ldots$ are i.i.d. (0, 1) Bernulli variables and Y is the result of flipping a coin, and we define the process conditional on $Y$ $$ X(n) = \begin{cases} \sum_{i=1}^n b_i& \text{if } Y = H\\ \sum_{i=1}^n -b_i& \text{if } Y = T \end{cases} $$ Then this process is not ergodic because initially it can take any integer value but, once the coin is flipped, it can take only positive or only negative values. That is, once you flip the coin the probability of reaching some states becomes zero. Therefore, the the ensamble average changes with $Y$.

Another loose way of thinking at ergodicity is to say that if any two collections of random variables partitioned far apart in the sequence are essentially independent.

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