Stationary covariance bound for reversible Markov chains
= Stationary covariance bound for reversible Markov chains
If $\|P\|_{L^2_0(\mu)}\leq r<1$, a stationary <reversible Markov chain> satisfies
$$
|\operatorname{Cov}(f(X_0),f(X_k))|\leq r^k\operatorname{Var}_\mu(f).
$$
Consequently the <variance> of a sum of $N$ consecutive values is at most $N(1+r)/(1-r)$ times the stationary <variance> of one value. This bounds the <Markov chain Monte Carlo asymptotic variance> uniformly over square-integrable functions.