Concentration of Markov chains with bounded moments

Let $\{W_{t}\}_{t=1}^{\infty }$ be a finite state stationary Markov chain, and suppose that $f$ is a real-valued function on the state space. If $f$ is bounded, then Gillman’s expander Chernoff bound (1993) provides concentration estimates for the random variable $f(W_{1})+\cdots +f(W_{n})$ that depend on the spectral gap of the Markov chain and the assumed bound on $f$. Here we obtain analogous inequalities assuming only that the $q$’th moment of $f$ is bounded for some $q\geq 2$. Our proof relies on reasoning that differs substantially from the proofs of Gillman’s theorem that are available in the literature, and it generalizes to yield dimension-independent bounds for mappings $f$ that take values in an $L_{p}(\mu )$ for some $p\geq 2$, thus answering (even in the Hilbertian special case $p=2$) a question of Kargin (Ann. Appl. Probab.17 (4) (2007) 1202–1221).