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Asymptotics for the finite time ruin probability in the renewal model with consistent variation. (English) Zbl 1130.60312
Summary: This paper investigates the finite time ruin probability in the renewal risk model. Under some mild assumptions on the tail probabilities of the claim size and of the inter-occurrence time, a simple asymptotic relation is established as the initial surplus increases. In particular, this asymptotic relation is requested to hold uniformly for the horizon varying in a relevant infinite interval. The uniformity allows us to consider that the horizon flexibly varies as a function of the initial surplus, or to change the horizon into any nonnegative random variable as long as it is independent of the risk system.

MSC:
60K05 Renewal theory
60F10 Large deviations
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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