Decoupling for Markov Chains

Abstract

Consider a Markov chain $(X_i)_{i\ge0}$ with invariant measure $\mu$ that admits the representation $X_{i+1}=\Phi(X_i,U_i)$, where $(U_i)_{i\ge0}$ are i.i.d. random variables and $\Phi$ is a measurable map. We introduce a tangent-decoupled process $(\widetilde X_i)_{i\ge0}$ obtained by replacing $(U_i)$ with an independent copy. Conditional on the realized backbone $(X_i)$, the sequence $(f(\widetilde X_i))$ is independent. Although $(\widetilde X_i)$ is not Markovian, under the same ergodicity assumptions that ensure a law of large numbers for $(X_i)$, the empirical averages $n^{-1}\sum_{i=1}^n f(\widetilde X_i)$ converge almost surely to $\mu(f)$. In addition, for every $f\in L^2(\mu)$ and every $N\ge1$, $$ \operatorname{Var}\!\Bigl(\sum_{i=1}^N f(X_i)\Bigr) \;\le\; 2\,\operatorname{Var}\!\Bigl(\sum_{i=1}^N f(\widetilde X_i)\Bigr), $$ and therefore $\sigma_f^2 \le 2\,\widetilde\sigma_f^{\,2}$ for the corresponding time-average variance constants. The inequality requires neither reversibility nor mixing assumptions. Its proof identifies the two sequences as tangent in the sense of decoupling theory and applies the sharp $L^2$ tangent decoupling inequality of de la Pe\~na, Yao, and Alemayehu (2025).