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Collective risk theory is concerned with random fluctuations of the total net assets, the risk reserve, of an insurance company. In this paper we consider weak approximations in risk theory which are especially relevant whenever the claim experience allows for heavy-tailed claims. We approximate the risk process by an α-stable Lévy motion (1 < α < 2) with drift. The ruin probability within a finite time horizon is estimated. Finally, a numerical example is presented.