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(ξ 1 + · · · + ξ η > 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.

No abstract

No abstract

No abstract

No abstract