Orderings of risks: A comparative study via stop-loss transforms

The relations between the following concepts for ordering risks are investigated: stochastic dominance, stop-loss order, convex order and being more dangerous. Using characterizations via stop-loss transforms, we give an elementary proof of the separation theorem for stop-loss order, and we correct a mistake in a result of van Heerwaarden (1991) about the connection between stop-loss order and being more dangerous. This is done by introducing a new notion of convergence for distributions. Moreover, we consider lattice properties of these orders.