Capital allocation rules and generalized collapse to the mean

In the context of capital allocation principles for (not necessarily coherent) risk measures, we derive - under mild conditions - some representation results as ''collapse to the mean" in a generalized sense. This approach is related to the well-known Gradient allocation and allows to extend a result of Kalkbrener (Theorem 4.3 in [27]) to a non-differentiable setting as well as to more general capital allocation rules and risk measures.