We establish preservation results for the stochastic comparison of multivariate random sums of stationary, not necessarily independent, sequences of nonnegative random variables. We consider convex-type orderings, i.e. convex, coordinatewise convex, upper orthant convex and directionally convex orderings. Our theorems generalize the well-known results for the stochastic ordering of random sums of independent random variables. Introduction. In this paper we establish preservation results for the

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... stochastic comparison of multivariate random sums of nonnegative random variables. We consider convex, coordinatewise convex, upper orthant convex and directionally convex orderings. The first three are variability orderings, whereas the latter can be considered as a variability-dependence ordering. We refer the reader to the books [7] or [10] . Using so called supermodular functions we are able to extend the results known in the case of independent random variables ([2], [8], [9]) to stationary sequences. Random sums are used in many applied sciences, and comparison of such sums plays an important role in stochastic models, where exact calculations of some quantities are not possible. For example in the context of multivariate shock models preservation of the stochastic ordering for (multivariate) random sums was established in [8] and [12]. In actuarial science random sums were considered in [1] , and in the context of queueing systems in [2] and [4]. In the latter paper a preservation result was proved (in the directionally convex case) not only for random sums but for more general functionals of stationary point processes. The paper is organized as follows. In Section 2 we collect some needed definitions and technical lemmas. In Section 3 we prove regularity properties