Moment bounds for dependent sequences in smooth Banach spaces

We prove a Marcinkiewicz-Zygmund type inequality for random variables taking values in a smooth Banach space. Next, we obtain some sharp concentration inequalities for the empirical measure of $\{T, T^2, \cdots, T^n\}$, on a class of smooth functions, when $T$ belongs to a class of nonuniformly expanding maps of the unit interval.