  The aim of this paper is to propose new Rosenthal-type inequalities for
moments of order higher than 2 of the maximum of partial sums of stationary
sequences including martingales and their generalizations. As in the recent
results by Peligrad et al. [Proc. Amer. Math. Soc. 135 (2007) 541-550] and Rio
[J. Theoret. Probab. 22 (2009) 146-163], the estimates of the moments are
expressed in terms of the norms of projections of partial sums. The proofs of
the results are essentially based on a new maximal inequality generalizing the
Doob maximal inequality for martingales and dyadic induction. Various
applications are also provided.
