  We investigate a new natural class $\mathcal{J}$ of probability distributions
modeling large claim sizes, motivated by the `principle of one big jump'.
Though significantly more general than the (sub-)class of subexponential
distributions $\mathcal{S}$, many important and desirable structural properties
can still be derived. We establish relations to many other important large
claim distribution classes (such as $\mathcal{D}$, $\mathcal{S}$,
$\mathcal{L}$, $\mathcal {K}$, $\mathcal{OS}$ and $\mathcal{OL}$), discuss the
stability of $\mathcal{J}$ under tail-equivalence, convolution, convolution
roots, random sums and mixture, and then apply these results to derive a
partial analogue of the famous Pakes-Veraverbeke-Embrechts theorem from ruin
theory for $\mathcal{J}$. Finally, we discuss the (weak) tail-equivalence of
infinitely-divisible distributions in $\mathcal{J}$ with their L\'{e}vy
measure.
