LAWS OF LARGE NUMBERS AND CENTRAL LIMIT THEOREMS FOR RANDOM GEOMETRIC MEASURES. A COMPUTATIONAL APPROACH
type: CreativeWork
description: 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 possib…
abstract: 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 possib…
author: I. Matuonto
language: eng
keywords: Settore MAT/06 - Probabilita' e Statistica Matematica
date published: 2010-12-17
date modified: 2026-07-27T20:19:17Z
provider: Università degli Studi di Milano
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Università degli Studi di Milano

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value: http://hdl.handle.net/2434/150202