A (discrete) memoryless information source is (usually) defined as a collection of random variables that are independent and identically distributed. My question is, does memorylessness imply stationarity and ergodicity? What about the converse?
2026-03-30 12:45:13.1774874713
Stationarity and Ergodicity vs. Memorylessness
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In this sense, memorylesssness of $\left(X_n\right)_{n\geqslant 1}$ implies its stationarity and ergodicity. Ergodicity follows from Kolmogorov's $0$-$1$ law and stationarity by using the fact that the law of $(X_1,\dots,X_n)$ is the product of the common law of $X_n$'s.
The converse is not true: if $\left(\varepsilon_i\right)_{i\in\mathbb Z}$ is an i.i.d. sequence, then $(X_n):=(\varepsilon_n+\varepsilon_{n+1})$ is strictly stationary, ergodic but in general not independent.