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Moment inequalities and complete moment convergence. (English) Zbl 1180.60019
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 Y. S. Chow [Bull. Inst. Math., Acad. Sin. 16, No. 3, 177–201 (1988; Zbl 0655.60028)], M.-H. Zhu [Discrete Dyn. Nat. Soc. 2007, Article ID 74296 (2007; Zbl 1181.60044)], and Y.-F. Wu and 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.

MSC:
60E15 Inequalities; stochastic orderings
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