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In this paper, we discuss max-sum equivalence and convolution closure of heavy-tailed distributions. We generalize the well-known max-sum equivalence and convolution closure in the class of regular variation to two larger classes of heavy-tailed distributions. As applications of these results, we study asymptotic behaviour of the tails of compound geometric convolutions, the ruin probability in the compound Poisson risk process perturbed by an α-stable Lévy motion, and the equilibrium waiting-time distribution of the M/G/k queue.