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The aim of this paper is to propose new Rosenthal-type inequalities formoments of order higher than 2 of the maximum of partial sums of stationarysequences including martingales and their generalizations. As in the recentresults 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 areexpressed in terms of the norms of projections of partial sums. The proofs ofthe results are essentially based on a new maximal inequality generalizing theDoob maximal inequality for martingales and dyadic induction. Variousapplications are also provided.

Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples