This dissertation is composed by two blocks.

The first part is concerned with several types of distributional approximations, namely multivariate Poisson, Poisson process and Gaussian approximation.

Employing the solution of the Stein equation for Poisson distribution, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes, to provide a Poisson process approximation result in terms of a new metric called dπ, defined as the supremum overall Wasserstein distances between random vectors obtained evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure and the Poisson process approximation of point processes with Papangelou intensity are considered.

Next, we consider a variant of the classical Johnson Mehl birth-growth model with random growth speed and prove Gaussian approximation results. In this model, seeds appear at random times and locations and start growing instantaneously in all directions with random speeds. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed and birth
time distribution, we establish quantitative central limit theorems for the sum of given weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with deterministic growth speed.

In the second part of the dissertation, we propose general construction of convex closed sets obtained by applying sublinear expectations to random vectors in Euclidean space. We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of our construction. Further, we identify the dual representation of such convex bodies and identify one map that serves as a building block for all so defined convex bodies. Several further properties are investigated.