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Approximations for compound Poisson and Pólya processes. (English) Zbl 0576.62098
Consider the compound distribution \(G(y)=P\{\sum^{N}_{1}Y_ k\leq y\}\) where \(\{Y_ k\), \(k\in {\mathbb{N}}\}\) is a sequence of i.i.d. non- negative r.v. and N an integer r.v. independent of \(\{Y_ k\), \(k\in {\mathbb{N}}\}\). For example G may represent the probability distribution of a portfolio in an insurance company where the process \(\{Y_ k\), \(k\in {\mathbb{N}}\}\) represents the successive claims while N denotes the number of claims.
For N either Poisson or a Pólya process we derive asymptotic saddle- point expansions for 1-G(y) when y becomes large. The conditions and proofs for both cases are somewhat different. Numerical computation for uniform and exponential distributions show the accuracy of the obtained approximations; a comparison with the classical Esscher approximation is also provided.
Reviewer: J.L.Teugels

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
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