Theory of Probability &Its Applications, Volume 65, Issue 1, Page 1-16, January 2020.
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.

中文翻译：

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线性和非线性统计量的矩不等式

概率论及其应用，第65卷，第1期，第1-16页，2020年1月。
我们考虑以下形式的统计信息：$ T = \ sum_ {j = 1} ^ n \ xi_ {j} f_ {j} + \ mathcal R $，其中$ \ xi_j，f_j $，$ j = 1，\ dots，n $和$ \ mathcal R $是$ \ mathfrak M $某些$ \ sigma $-代数$ \ mathfrak M的可测量随机变量$。假设存在$ \ sigma $-代数$ \ mathfrak M ^ {（（n）} $，$ \ mathfrak M ^ {{j}} \ subset \ mathfrak M $，$ j = 1，\ dots，n $，这样$ {E} {（\ xi_j \ mid \ mathfrak M ^ {（j）}）} = 0 $。在这些假设下，我们证明了$ {E} | T | ^ p $与$ p \ ge 2 $的不等式。我们还将讨论本文主要结果在估计线性形式矩，$ U $统计量以及维格纳半圆定律的Stieltjes变换的特征方程式摄动中的应用。