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

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 + ... + 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.