Poissonovi točkovni procesi

master's thesis

• Author: Andrijana Brkić
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Zvonimir Tutek (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu, cilj nam je bio definirati i dati osnovne rezultate o Poissonovim točkovnim procesima. Prije svega, prikazana je definicija općenitih točkovnih procesa za koju se tradicionalno koriste dva slična, ali ipak različita pristupa - prvi se koncentrira direktno na točke kao podskupove od \(\mathbb{R}^d\), a drugi uvodi pojam slučajnih mjera te na njima gradi definiciju. Iako se uglavnom oslanjamo na prvu definiciju, naglašavamo neke zanimljive razlike i veze sa drugom definicijom. Nakon formalne definicije točkovnih procesa, a onda i konkretno Poissonovih točkovnih procesa, pokazujemo osnovne rezultate vezane uz transformacije Poissonovih procesa poput unije prebrojivo procesa (teorem superpozicije), preslikavanja, stanjivanja i označavanja procesa. U zadnjem poglavlju dokazana su još dva bitna rezultata, a to su Campbellov teorem koji opisuje distribuciju slučajne varijable \[\sum_{X \in \varPi}f(X),\] te Rényijev teorem koji daje vrlo zanimljivu karakterizaciju Poissonovih procesa kroz funkciju izbjegavanja.

Abstract (english)

In this thesis, our goal was to define Poisson point processes and introduce some of their basic properties. Firstly, we reviewed the definition of general point processes. Typically, two similar, but different approaches are used for the definition - the first one concentrates directly on the points as subsets of \(\mathbb{R}^d\), while the second approach relies on the concept of random measures. Although we concentrate mainly on the first definition, we point out interesting differences and similarities between those two approaches. After the formal definition of point processes and Poisson point processes, we show some fundamental results from the transformation theory of Poisson processes concerning countable unions of processes (superposition theorem), mapping, thinning and marking. In the last chapter we prove two more substantial results: the Campbell theorem which describes the distribution of the random variable \[\sum_{X \in \varPi}f(X),\] and Rényi's theorem which reveals an interesting characterisation of Poisson point processes using the avoidance function.

• Keywords: Poissonovi točkovni procesi; općeniti točkovni procesi; transformacije Poissonovih procesa; Campbellov teorem; Rényijev teorem
• Keywords (english): Poisson point processes; general point processes; transformation theory of Poisson processes; Campbell theorem; Rényi's theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:148169
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-06-08 12:59:26