Moment inequalities for linear and nonlinear statistics

We consider statistics of the form $T =\sum_{j=1}^n \xi_{j} f_{j}+ \mathcal R $, where $\xi_j, f_j$, $j=1, \dots, n$, and $\mathcal R$ are $\mathfrak M$-measurable random variables for some $\sigma$-algebra $ \mathfrak M$. Assume that there exist $\sigma$-algebras $\mathfrak M^{(1)}, \dots, \mathfrak M^{(n)}$, $ \mathfrak M^{(j)} \subset \mathfrak M$, $j=1, \dots, n$, such that ${E}{(\xi_j\mid \mathfrak M^{(j)})}=0$. Under these assumptions, we prove an inequality for ${E}|T|^p$ with $p \ge 2$. We also discuss applications of the main result of the paper to estimation of moments of linear forms, $U$-statistics, and perturbations of the characteristic equation for the Stieltjes transform of Wigner's semicircle law.