Moment inequalities and complete moment convergence
Let \(\{Y_i\), \(1\leq i\leq n\}\) and \(\{Z_i\), \(1\leq i\leq n\}\) be sequences of random variables. For any \(\varepsilon> 0\) and \(a> 0\), bounds for \[ E\Biggl(\Biggl| \sum^n_{i=1} (Y_i+ Z_i)\Biggr|-\varepsilon a\Biggr)^+ \] and \[ E\Biggl(\max_{1\leq k\leq n}\Biggl|\sum^k_{i= 1} (Y_i+ Z_i)\Biggr|-\varepsilon a\Biggr)^+ \] are obtained. From these results, we establish general methods for obtaining the complete moment convergence. The results of \textit{Y. S. Chow} [Bull. Inst. Math., Acad. Sin. 16, No.~3, 177--201 (1988; Zbl 0655.60028)], \textit{M.-H. Zhu} [Discrete Dyn. Nat. Soc. 2007, Article ID 74296 (2007; Zbl 1181.60044)], and \textit{Y.-F. Wu} and \textit{D. Zhu} [J. Korean Stat. Soc., in press (2009)] are generalized and extended from independent (or dependent) random variables to random variables satisfying some mild conditions. Some applications to dependent random variables are discussed.
scientific article; zbMATH DE number 5624530