Elementary renewal theorems for widely dependent random variables with applications to precise large deviations

Abstract

On the basis of Wang and Cheng (J. Math. Anal. Appl. 384 (2011) 597–606), this paper further investigates elementary renewal theorems for counting processes generated by random walks with widely orthant dependent increments. The obtained results improve the corresponding ones of the above-mentioned paper mainly in the sense of weakening the moment conditions on the positive parts of the increments. Meanwhile, a revised version of strong law of large numbers for random walks with widely orthant dependent increments is established, which improves Theorem 1.4 of Wang and Cheng (2011) by enlarging the regions of dominating coefficients. Finally, by using the above results, some precise large deviation results for a nonstandard renewal risk model are established, in which the innovations are widely orthant dependent random variables with common heavy tails, and the inter-arrival times are also widely orthant dependent.

Keywords:

• Elementary renewal theorems
• counting processes
• wide dependence
• moment conditions
• precise large deviations

2000 Mathematics Subject Classification:

• Primary 60F05
• Secondly 60F25

Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments on the earlier version of this paper.

Additional information

Funding

Research supported by National Natural Science Foundation of China (No. 11401415). This paper was also supported by Jiangsu Overseas Research and Training Program for Prominent University Young and Middle-aged Teachers and Presidents.