Long-run variance for a stationary Markov chain

Let $(X_k{)}_{k\ge 0}$ be a stationary time-homogeneous Markov chain with invariant probability $\pi$, and let $f$ be real Borel with $\pi (f)=0$ and $\pi (f^2)<\mathrm{\infty }$. For $k\ge 0$, set ${\gamma }_k:={\mathrm{Cov}}_{\pi }(f(X_0),f(X_k))$. When the Markov-chain CLT $n^{-1\mathrm{/}2}{\sum }_{k=0}^{n-1}f(X_k)\Rightarrow N(0,{\sigma }_f^2)$ holds with finite variance parameter and ${\sum }_{k\ge 1}\mathrm{\mid }{\gamma }_k\mathrm{\mid }<\mathrm{\infty }$, the covariance long-run variance is ${\sigma }_{\mathrm{L}\mathrm{R}}^2(f):={\gamma }_0+2{\sum }_{k=1}^{\mathrm{\infty }}{\gamma }_k\mathrm{.}$

This definition does not infer a CLT from covariance summability, or infer that a weak-limit variance equals the covariance sum. Whenever ${\sigma }_{\mathrm{L}\mathrm{R}}^2(f)$ is used as the variance parameter in a CLT or MCSE claim, that equality must be established by the theorem being applied. For example, the local Poisson-equation theorem proves the equality under its square-integrable solution hypotheses.

Under stationarity, the finite variance identity is $\frac{1}{n}{\mathrm{Var}}_{\pi }\text{\hspace{0.17em}⁣}({\sum }_{k=0}^{n-1}f(X_k))={\gamma }_0+2{\sum }_{k=1}^{n-1}(1-\frac{k}{n}){\gamma }_k\mathrm{.}$ Absolute summability makes the right side converge to ${\sigma }_{\mathrm{L}\mathrm{R}}^2(f)$ by dominated convergence. Equality between this variance limit and a variance parameter from weak convergence is a separate assertion; it is not part of this definition.