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Criteria for the stochastic ordering of random sums, with actuarial applications. (English) Zbl 1003.60022
Let \({\mathbf M}=(M_1,\ldots,M_{n})\) and \({\mathbf M'}=(M_1',\ldots,M_{n}')\) be random vectors of integer-valued random variables and let \({\mathbf M}\) and \({\mathbf M'}\) have the same univariate marginals but different joint distributions. For given sequences \((X_{1k}),\ldots,(X_{nk})\) and \((X_{1k}'),\ldots,(X_{nk}')\) of positive random variables let us consider the sums \[ S_{M_{i}}=\sum_{k=1}^{M_{i}}X_{ik},\quad S_{M_i'}=\sum_{k=1}^{M_i'}X_{ik}', \qquad 1\leq i\leq n. \] The sequences \((X_{ik})\) and \((X_{ik}')\) are identically distributed and independent of \({\mathbf M}\) and \({\mathbf M'}\). The sequences \((X_{ik})\) and \((X_{i'k})\) are independent for \(i\neq i'\), but for a fixed \(i\), no assumption is made on the dependence structure between elements of any particular sequence \((X_{ik})\). The authors give definitions of the upper orthant, the lower orthant, the concordance, and the supermodular ordering and prove that vectors of sums \((S_{M_1},\ldots,S_{M_{n}})\) and \((S_{M_1'}',\ldots,S_{M_{n}'}')\) are stochastically ordered in just the same way as the corresponding vectors \({\mathbf M}\) and \({\mathbf M'}\). Some applications of the obtained results to the collective risk model in actuarial science in which a portfolio consists of \(n\geq 2\) classes of business whose respective numbers of claims are random variables \(M_1,\ldots, M_{n}\) are presented.

MSC:
60E15 Inequalities; stochastic orderings
91B30 Risk theory, insurance (MSC2010)
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