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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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