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,T2,⋯,Tn}{T,T2,⋯,Tn}, on a class of smooth functions, when TT belongs to a class of nonuniformly expanding maps of the unit interval.