On the asymptotic approximation of inverse moment for non negative random variables

ABSTRACT

In this paper, we have studied the asymptotic approximation of inverse moment. Let {Z_n} be non negative random variables, where the truncated random variables satisfy the Rosenthal-type inequality. Denote . The inverse moment can be asymptotically approximated by , and the growth rate is presented as . Meanwhile, we obtain a result for a function f(x) satisfying certain conditions. On the other hand, if , then the growth rate is presented as |E(a + X_n)^{− α}/(a + EX_n)^{− α} − 1| = O(1/EX_n). Our results generalize and improve some corresponding ones. Finally, some examples of inverse moment are illustrated.

KEYWORDS:

• Asymptotic approximation
• Growth rate
• Inverse moment
• Non negative random variables.

Mathematical Subject Classification:

• 60E15
• 62G20

Acknowledgments

The authors are deeply grateful to the Editor-in-Chief Prof. N. Balakrishnan, anonymous associate editor, and the referees whose insightful comments and suggestions have contributed substantially to the improvement of this paper.

Funding

This work is supported by National Natural Science Foundation of China (11501005, 11671012), National Social Science Fund of China (14ATJ005), Natural Science Foundation of Anhui Province (1508085J06, 1608085QA02) and Provincial Natural Science Research Project of Anhui Colleges (KJ2014A020, KJ2015A065, KJ2016A027, KJ2017A027).