Abstract
In this paper, we find the upper bound for the tail probability Psupn⩾0∑I=1nξI>x with random summands ξ1,ξ2,… having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ1exp{−ϱ2x} with some positive constants ϱ1 and ϱ2. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.
Highlights
Let {ξ 1, ξ 2, . . .} be a sequence of independent and real-valued random variables (r.v.s)
In 1997, Sgibnev generalized results of Kiefer and Wolfowitz [1] by obtaining the upper bound for submultiplicative moment
The main object of our research is the ruin probability of the renewal risk model
Summary
Let {ξ 1 , ξ 2 , . . .} be a sequence of independent and real-valued random variables (r.v.s). In the case of exponential function φ, Theorem 1 implies the following upper estimation for the tail probability of r.v. M∞. The assertion of Theorem 3 provides an algorithm to calculate two positive constants controlling the exponential upper bound. The derived upper exponential-type estimates for P(Sn > x, C) “work” under quite restrictive requirements for summands of sum Sn. The main object of our research is the ruin probability of the renewal risk model. In order to obtain a good and general upper bound of this probability, we use the estimate of probability P(max16k6n Sk > x ) presented in Lemma 1 In this lemma the requirements for summands of Sn are minimal
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