  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 $p\ge 2$, thus answering (even in the
Hilbertian special case $p=2$) a question of Kargin (2007).
