## 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:

 6e+16 Inequalities; stochastic orderings

### Citations:

Zbl 0655.60028; Zbl 1181.60044
Full Text:

### References:

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