We prove that an irreducible aperiodic Markov chain is geometrically ergodic if and only if any separately bounded functional of the stationary chain satisfies an appropriate subgaussian deviation inequality from its mean.
"Subgaussian concentration inequalities for geometrically ergodic Markov chains." Electron. Commun. Probab. 20 1 - 12, 2015. https://doi.org/10.1214/ECP.v20-3966