Moment inequalities for sums of dependent multiindexed random variables

Upper estimates of absolute moments are established, in case of a centered random field on a lattice $\mathbf{Z}^d$ ($d\ge1$), for sums over finite sets of arbitrary configuration. The dependence condition is given in a form of inequalities for covariances of certain powers of the initial random variables. It is shown that this condition can be deduced, under moment conditions on summands, from the usual mixing conditions for the field as well as from assumption that the field is either positively or negatively dependent.