Ergodicity

In a word, ergodicity means that averaging over time and averaging over ensembles is the same thing. That is, consider (i) a single realization of a random process, $$x_i[n]$$, and (ii) the random process $$x_1[n_0],x_2[n_0], \dots$$ that contains an infinite number of random variables obtained by picking the timepoint $$n_0$$ fro all the realizations $$x_1, x_2, \dots$$. Then, ergodicity says that the two aforementioned means will be the same, i.e.,

$$\lim_{M\to \infty}\frac{1}{M}\sum_{m=1}^M x_m[n_0] = \lim_{N\to \infty} \sum_{n=1}^N x_i[n]$$