In this paper, we consider the random sums of i.i.d. random variables ξ1,ξ2,... with consistent variation. Asymptotic behavior of the tail P(ξ1 + ... + ξη > x), where η is independent of ξ1,ξ2,..., is obtained for different cases of the interrelationships between the tails of ξ1 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.
Random sums Consistent variation Compound renewal risk model Ruin probability
Leipus, R., Šiaulys, J.: Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes. Insur.: Math. Econ. 40, 498–508 (2007)MATHCrossRefGoogle Scholar
Mikosch, T.: Non-Life Insurance Mathematics. Springer-Verlag, New York (2004)MATHGoogle Scholar
Ng, K.W., Tang, Q.H., Yan, J.-A., Yang, H.: Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Probab. 41, 93–107 (2004)MATHCrossRefMathSciNetGoogle Scholar