Strict stationarity is preserved by a measurable functional

⚠️ DEPRECATED — this statement is FALSE as written

Do not attempt to prove this theorem: it has a counterexample. Deprecated by its submitter (LukeBernese). Use refMarkovChainCLT.isStrictlyStationary_comp_of_measurable instead, which adds the missing hypothesis hY : ∀ n, Measurable (Y n) and is proved.

Why it is false. IsStrictlyStationary is an equality of pushforward measures, and the pushforward along a non-measurable map is the zero measure by convention. So when $Y$ is not measurable the hypothesis degenerates to $0 = 0$ and carries no information, while the conclusion can still be a substantive claim — because $g \circ Y_i$ may be measurable even when $Y_i$ is not.

Explicit counterexample. Take $\mathsf{X} = \mathbb{R} \times \{0,1\}$ and $Y_i = (h, b_i)$ with $h$ non-measurable and each $b_i$ measurable, and let $g = \mathrm{snd}$. Then each $Y_i$ is non-measurable, so the stationarity hypothesis holds vacuously, while $g \circ Y_i = b_i$ is measurable; choosing $\{b_i\}$ non-stationary (e.g. $b_0 \equiv 0$ and $b_i \equiv 1$ for $i \ge 1$) makes the conclusion false.

Assuming each $Y_n$ measurable — which holds in every application, where $Y$ is the coordinate process of a path space — makes the pushforward identities available and the result true.

If a sequence $Y = \{Y_i\}_{i \ge 0}$ of random elements is strictly stationary and $g$ is measurable, then the functional process $\{g(Y_i)\}_{i \ge 0}$ is strictly stationary.

Strict stationarity is the statement that the law of the whole shifted sequence $(Y_{k}, Y_{k+1}, \dots)$ on the path space does not depend on $k$. Applying the measurable map $(x_0, x_1, \dots) \mapsto (g(x_0), g(x_1), \dots)$ to both sides and using functoriality of the pushforward gives the claim.

This is the routine bookkeeping step needed to feed the functional process $\{f(X_n)\}$ of a stationary Markov chain into a central limit theorem stated for general stationary sequences, as in Jones's derivation of Corollaries 1 and 3-5.