Decoupling for Markov Chains

TL;DR

This paper introduces a tangent-decoupled process for Markov chains, demonstrating convergence of empirical averages and establishing variance bounds without requiring reversibility or mixing assumptions.

Contribution

It presents a novel tangent-decoupled process for Markov chains and proves convergence and variance inequalities using decoupling theory, extending analysis beyond traditional assumptions.

Findings

Empirical averages of the decoupled process converge almost surely.

Variance of the original process is bounded by twice that of the decoupled process.

The approach does not require reversibility or mixing assumptions.

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).