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.

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We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

Let {ξ 1 , ξ 2 , . . .} be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability P(sup n 0 n i=1 ξ i > x) can be bounded above by 1 exp{− 2 x} with some positive constants 1 and 2 . A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.