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Large deviations for heavy-tailed random sums in compound renewal model. (English) Zbl 0977.60034
The object of interest is a doubly stochastic sum \(S_t = \sum_{i=1}^{N(t)} X_i\), where \(N(t)\) is a random variable with finite expectation \(\lambda(t)\), and \(\lambda(t)\) goes to infinity for \(t \to \infty\), and the random variables \(X_i\) are of extended regular variation. The authors determine the tail behaviour of this sum, and relax the assumptions of a theorem of Klüppelberg and Mikosch who considered the same question. Applications to compound renewal models are given.

MSC:
60F10 Large deviations
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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