Concentration of Markov chains with bounded moments

Let {Wt}∞t=1 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 (W1) + · · · +f (Wn) 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 ≥ 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 Lp(μ) for some p ≥ 2, thus answering (even in the Hilbertian special case p = 2) a question of Kargin (Ann. Appl. Probab. 17 (4) (2007) 1202-1221).