Article

Extremes

, Volume 11, Issue 3, pp 261-279

Tail behavior of random sums under consistent variation with applications to the compound renewal risk model

  • Aldona AleškevičienėAffiliated withInstitute of Mathematics and Informatics
  • , Remigijus LeipusAffiliated withDepartment of Mathematics and Informatics, Vilnius UniversityInstitute of Mathematics and Informatics Email author 
  • , Jonas ŠiaulysAffiliated withDepartment of Mathematics and Informatics, Vilnius UniversityInstitute of Mathematics and Informatics

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Abstract

In this paper, we consider the random sums of i.i.d. random variables ξ 1,ξ 2,... with consistent variation. Asymptotic behavior of the tail P(ξ1 + ... + ξη > x), where η is independent of ξ 1,ξ 2,..., is obtained for different cases of the interrelationships between the tails of ξ 1 and η. Applications to the asymptotic behavior of the finite-time ruin probability ψ(x,t) in a compound renewal risk model, earlier introduced by Tang et al. (Stat Probab Lett 52, 91–100 (2001)), are given. The asymptotic relations, as initial capital x increases, hold uniformly for t in a corresponding region. These asymptotic results are illustrated in several examples.

Keywords

Random sums Consistent variation Compound renewal risk model Ruin probability

AMS 2000 Subject Classifications

Primary—60G50; Secondary—60F10, 62P05