LAWS OF LARGE NUMBERS AND CENTRAL LIMIT THEOREMS FOR RANDOM GEOMETRIC MEASURES. A COMPUTATIONAL APPROACH
The aim of this thesis is to produce computational experiments related to a Central Limit Theorem (CLT) for random measures proved by Penrose in [4]. In particular we analyze some random measures associated with a segment process, a Voronoi tessellation and a Johnson-Mehl tessellation constructed on an underlying 2-dimensional point process. In the result by Penrose, this point process is supposed to be Poisson non-homogeneous (or Binomial, but we are not going to study this case). In this thesis, we also consider the case of an underlying Neyman-Scott cluster process. The main references are [4] and references therein; before this CLT, in [5] Penrose obtained a Law of Large Numbers for random measures. These results are an extension of [1], where Baryshnikov and Yukich proved similar theorems under stricter assumptions on the radius of stabilization of the considered random measure. There are also other results concerning CLTs on random measures(for example in [2]), but processes are always assumed to be homogeneous; through Baryshnikov, Yukich and Penrose’s approach it is possible to relax the homogeneity conditions, so that their hypotheses are more likely to be true for applications. The main difference between other CLTs for random measures and this new approach is that the limit is not considered as the window of observation increases (which is a problem, because we should have homogeneity conditions of the random process on the whole space), but as the intensity of the process increases (just one realization of a process with sufficiently large intensity gives information on the typical features of the process). In Baryshnikov, Yukich and Penrose’s works new theoretical results and some examples are presented, but simulations are not included; moreover, the hypothesis of Poisson distribution of the underlying point process is extended only to the case of a Binomial point process. This thesis is a computational approach to the theorem, which may lead to possible future generalizations from a theoretical point of view, including more general point processes, such as the cluster ones. In our work we analyze the behaviour of some particular random measures as the intensity of the underlying point process increases. We find some advantages and disadvantages of Penrose’s approach; we have a validation of the theorem through our simulations by observing a Gaussian trend of our data (and by 2 performing some Lilliefors tests), and we also obtain some Laws of Large Numbers. Our computational approach has been carried out for measures depending on homogeneous Poisson point processes, non-homogeneous ones and Neyman-Scott ones. References [1] Yu Baryshnikov, J.E.Yukich, Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 2005, 213-253. [2] V. Benes, J. Rataj, Stochastic Geometry: Selected Topics. Kluwer Academic Publishers, Boston, 2004.[3] V.Capasso, M.Burger, A.Micheletti, C.Salani, Mathematical models for polymer crystallization processes. In Mathematical Modelling for Polymer Processing (V. Capasso, Ed). Springer, Heidelberg, 2000, 167-242. [4] M. D. Penrose, Gaussian limits for random geometric measures. Electronic Journal of Probability 12, 2007, 989-1035. [5] M. D. Penrose, Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 2007, 1124-1150. 4
Settore MAT/06 - Probabilita' e Statistica Matematica
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