University of Zagreb. Faculty of Science. Department of Mathematics.
Abstract
U ovom radu cilj je bio prikazati Poissonove točkovne procese. Na početku su definirane Poissonova vjerojatnosna distribucija i točkovni procesi općenito. Točkovne procese definiramo kao brojeću mjeru definiranu na slučajnom rasporedu točaka u nekom prostoru koja podskupu prostora pridružuje broj točaka koji upada u taj podskup. Osim definicije samog procesa, uvedeni su pojmovi mjere intenziteta i Laplaceova funkcionala. Navedene su i osnovne tvrdnje o točkovnim procesima i njihovoj distribuciji. U drugom poglavlju povezujemo Poissonovu distribuciju i točkovne procese u obliku Poissonovih procesa. Poissonov proces je točkovni proces kod kojeg slučajan broj točaka koji upada u dani skup ima Poissonovu distribuciju. Nakon same definicije, izveden je specifičan oblik Laplaceova funkcionala tog procesa te je dokazana egzistencija takvih procesa. Jedna od osnovnih karakterizacija Poissonovih procesa je dana u obliku Meckeove formule. Dokaz formule se provodi u nekoliko koraka i to je prikazano u trećem poglavlju. Posljednje poglavlje uvodi stacionarne točkovne procese te Palmovu distribuciju točkovnih procesa. Nakon same definicije stacionarnosti i Palmove distribucije, dokazan je Mecke-Slivnyakov teorem. To je varijanta Meckeove formule za stacionarne točkovne procese. Na kraju samog rada uvedeni su Voronoijev mozaik i formula inverzije pomoću kojih povezujemo stacionarnu distribuciju i Palmovu distribuciju točkovnih procesa.In this thesis, the aim was to present Poisson point processes. At first, we define the Poisson distribution and point processes in general. A point process is defined as a counting measure defined on a random set of points in some space which maps number of random points in subset of space to that subset. Besides the definition, we introduce the intensity measure and the Laplace functional of a point process. Also, some basic claims about point processes and their distribution are given in this section. In the second chapter we bring together the Poisson distribution and point processes in the form of the Poisson processes. A Poisson process is a point process for which the number of points in a given set has a Poisson distribution. After the definition, we derive the special form of the Laplace functional of a Poisson process and we prove the existence of these processes. One of the basic characterization of a Poisson process is the Mecke equation. Proof takes a few steps and these steps are given in the third chapter. Last chapter introduce the stationary point processes and the Palm distribution of a point processes. We prove the Mecke-Slivnyak theorem. That is the version of the Mecke equation for stationary point processes. In the end we introduce Voronoi tessellations and the inversion formula by which we derive connection between stationary distribution and the Palm distribution of the point processes
