I take great pleasure in addressing this audience. As you might know I'm a mathematician with a deep interest in insurance mathematics. As such, it is my sincere opinion that the gap between practising actuaries and theoretical researchers can be made substantially smaller. If my contribution can help in bridging the gap, I will feel fully compensated for the effort it took to prepare this lecture and the results contained therein.

The simple fact that we meet on the occasion of the sixteenth ASTIN Colloquium gives me a challenging opportunity to help in creating a platform on which both theoreticians and practitioners can meet.

The subject of my lecture stems from a long interest in large claims: What are they? Are they really dangerous? Is there a way to get them under control? Can one recognize them in practical situations?

I like to express in simple mathematical terms some results that might help in acquiring better insight on the impact of large claims in insurance mathematics. Perhaps, no result will be of immediate applicability as reality is too complicated to be described by the simplicity of the results to follow. Nevertheless the latter can be considered as building blocks of a real world in which one has to tackle large claims in theory as well as in practice.