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Large deviations of heavy-tailed sums with applications in insurance. (English) Zbl 0927.60037
Large deviation probabilities occur in a natural way in many applied areas, so for instance in some problems of insurance. The authors give a short review of results on large deviation probabilities for sums of i.i.d. random variables with special emphasis on distributions with heavy tails. Large deviation techniques are then applied to randomly indexed sums of i.i.d. random variables and Poisson shot noise processes. The paper also indicates the close relationship between large deviation results and the modeling of large insurance claims, in particular, reinsurance. Large deviation techniques are also applied to the estimation of the ruin probability in the Cramér-Lundberg model of risk theory.
##### MSC:
 60F10 Large deviations 60F05 Central limit and other weak theorems 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G70 Extreme value theory; extremal processes (probability theory) 60G50 Sums of independent random variables; random walks