Asymptotics for randomly weighted and stopped dependent sums

Abstract

This paper deals with the tail probability of , where is a sequence of extended negatively upper orthant dependent or bivariate upper tail independent, identically distributed random variables with dominatedly varying tails, is a sequence of nonnegative nondegenerate at zero random variables (not necessarily independent and identically distributed), is a random variable taking values in . In addition, , and are mutually independent. Under some mild conditions, the weak asymptotic equivalence relations for the probability are established. An application to the random-time ruin probability in the discrete time risk model is provided.

Keywords::

• asymptotic tail probability
• randomly weighted and stopped sums
• dominated variation
• extended negative upper orthant dependence
• bivariate upper tail independence
• random-time ruin probability

Keywords:

• 62E20
• 62P05

Acknowledgements

The authors thank the referees for their very useful comments and suggestions for improving the paper. The first author would like to thank the Faculty of Mathematics and Informatics, Vilnius University for its excellent hospitality.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China [No. 71471090], the Humanities and Social Sciences Foundation of the Ministry of Education of China [No. 14YJCZH182], the China Postdoctoral Science Foundation [No. 2014T70449, 2012M520964], the Natural Science Foundation of Jiangsu Province of China [No. BK20131339], Qing Lan Project, PAPD, Project of Construction for Superior Subjects of Statistics of Jiangsu Higher Education Institutions. The second and third authors are supported [by grant no. MIP-13079] from the Research Council of Lithuania. This work was finished during a visit of the first author to Vilnius University, supported by the Project ‘Competitive Financing of Short-Term Research Visits’, project code VP-3.1-SMM-01-V-02-001, by the Ministry of Education and Science of the Republic of Lithuania, Support Foundation European Social Fund Agency and the Research Council of Lithuania.