Heavy-tailed distributions have offered an active field of research for insurance mathematics, queuing theory and operations research for decades. This talk concentrates on phenomena that can be encountered in heavy-tailed models. Special emphasis is put on the so called principle of a single big jump. It means that the most likely way for the sum of i.i.d. variables to be large is that one of the summands itself is very large.

The simplest case where this principle can be observed is discussed in detail. To achieve this, we study the asymptotic properties of the variable \[ Z_d:=\frac{X_1}{d}\big{|}\{X_1+X_2=d\}, \] as $d\to \infty$. Here $X_1$ and $X_2$ are non-negative i.i.d. variables with a common twice differentiable density function $f$.

General results concerning the distributional limits of $Z_d$ are discussed with some examples. Eventual log-convexity or log-concavity of $f$ turns out to be the key ingredient that determines how the variable $Z_d$ behaves. As a consequence, two surprising discoveries are presented: Firstly, it is noted that the distributional limit is not strictly determined by the decay rate of the tail function.  

Secondly, it is shown that there exists a light-tailed distribution exhibiting behaviour that is commonly associated with heavy-tailed distributions i.e. the principle of a single big jump.