The problem under consideration started with a study of spacial change-point problem, or change-set problem as we called it: suppose there is a set $K$, such that our observations inside this set behave differently then anywhere outside it. Given $n$ observations, how can we test that a given $K$ is the correct one against its small perturbation $K(\varepsilon)$ as an alternative?
In huge amount of publications on the change-set problem, almost all devoted to the estimation of the change-set, we could not find such an object as an "alternative set" $K(\varepsilon)$. One reason for this porbably is that it is not easy to realize how to describe small changes in sets. We developed an appropriate notion of the derivative of set-valued function in (Khmaladze, 2007) and used it to build a version of contiguity theory in (Einmahl, Khmaladze, 2011) for statistical problems where the parameter of interest is a set.
One single result here is that if $\Phi_n$ is a point process with increasing intenstiy $n$, and the symmetric difference $K(\varepsilon)\Delta K$ shrinks and "vanishes" as $\varepsilon \to 0$, then the sequence $\Phi_n(K(\varepsilon)\Delta K)$ lives in the limit on the derivative set $dK(\varepsilon)/d\varepsilon$:
$$\Phi_n( K(\varepsilon)\Delta K) \to \Psi (dK(\varepsilon)/d\varepsilon, {\rm as} n\to\infty, \varepsilon\to 0.$$
In the talk we present the main framework of this approach. We hope that some discussions would lead to further applications.