Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions

Abstract

We obtain precise constants in the Marcinkiewicz-Zygmund inequality for martingales in $\mathbb{L}^{p}$ for p>2 and a new Rosenthal type inequality for stationary martingale differences for p in ]2,3]. The Rosenthal inequality is then extended to stationary and adapted sequences. As in Peligrad et al. (Proc. Am. Math. Soc. 135:541–550, [2007]), the bounds are expressed in terms of $\mathbb{L}^{p}$ -norms of conditional expectations with respect to an increasing field of sigma algebras. Some applications to a particular Markov chain are given.