Asymptotic tail probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks. Therefore, it is not surprising that the maxima represents the most influential factor when one investigates the tail behaviour of our considered risk aggregation, which, for example, can be found in the reinsurance market. This extreme behaviour confirms the ‘one big jump’ property that has been vastly discussed in the existing literature in various forms whenever asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate log-normal distribution.