This work uses geometric tools to develop new results in the theories of random sets, valuations, stochastic processes, and statistical testing. The first part is dedicated to studying the intersection of randomly translated sets. In the theory of random sets, the operations of union and of Minkowski sum have been thoroughly studied, as they agree well with the capacity functional of a random set, which defines its distribution. Then, limit theorems for random sets are mostly derived for their Minkowski sums and unions. Only recently, some limit theorems have been achieved for their intersections. This work expands on some of these results by using the asymptotic properties of integrals over Minkowski differences. The second contribution came out of necessity in the path of developing the third part of this thesis. The functions defined over the family of convex bodies which satisfy the additivity property, called valuations, are well-studied under assumptions of continuity and invariance under rigid motions. A complete characterisation of planar monotone integer-valued σ-continuous valuations taking integer values is presented, without assuming invariance under any group of transformations. A construction of the product for valuations of this type is introduced. In the third part of the thesis, a new family of set-indexed stochastic processes is presented. These processes satisfy the additivity property of valuations, so they are referred to as random valuations. The family of valuations is very rich. To achieve meaningful characterisation results, assumptions in the form of independence and infinite divisibility must be taken into consideration. Under these assumptions and by using tools from the theories of Lévy processes, stochastic geometry, and valuations, we are able to build a rich new theory, which is deeply connected with well-known results of deterministic valuations and integral geometry. The fourth contribution of this thesis is the development of new methods for testing a cone hypothesis about the mean of a Gaussian distribution, which can be expressed as a constraint testing problem. The proposed tests adapt based on the number of constraints which are violated. It improves the classical (non-adaptive) methods when few constraints are not satisfied, in terms of both simplicity and power. The new tests are shown to have a valid significance level α. Moreover, some possible tools to evaluate the elements of the family of adaptive tests are presented, in terms of risk and power.
Tommaso Visonà (Wed,) studied this question.