Elements of renewal theory for cluster point processes

doctoral thesis

• Title (croatian): Elementi teorije obnavljanja za točkovne procese s klasterima
• Author: Marina Dajaković
• Mentor: Bojan Basrak (mentor)
• Committee member: Nikola Sandrić (predsjednik povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Committee member: Alexander Marynych https://orcid.org/0000-0002-7628-7541 (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-11-10, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

The main goal of this thesis is to discuss renewal theorems in the framework of the cluster point processes on the real line. More precisely, under appropriate integrability assumptions, we describe the asymptotic distribution of the shifted renewal cluster point process and, as a consequence, we show that its mean measure on a bounded interval is asymptotically proportional to the length of the interval. Also, we prove the equivalent version of the latter result that allows one to determine the asymptotic behavior of the functions of special convolution form. These generalizations of the well-known results from the standard renewal theory are obtained using the methods and tools of modern probability theory, such as point process theory, especially vague convergence, and coupling method.

Abstract (croatian)

Glavni cilj ove disertacije je diskutirati teoreme obnavljanja u kontekstu točkovnih procesa s klasterima na skupu realnih brojeva. Preciznije, uz određene pretpostavke na prve momente, opisujemo asimptotsku distribuciju pomaknutog točkovnog procesa s klasterima te, kao posljedicu, pokazujemo da je srednja mjera takvog procesa na ograničenom intervalu proporcionalna duljini intervala. Također, dokazujemo ekvivalentnu verziju prethodnog rezultata koja omogućuje određivanje asimptotskog ponašanja konvolucije funkcija specijalnog oblika. Navedene generalizacije dobro poznatih rezultata iz standardne teorije obnavljanja su dobivene korištenjem metoda i alata moderne teorije vjerojatnosti, kao što su teorija točkovnih procesa, posebno vague konvergencija, te metoda sparivanja.

• Keywords: cluster point processes; Blackwell’s renewal theorem; the key renewal theorem; the extended renewal theorem
• Keywords (croatian): točkovni procesi s klasterima; Blackwellov teorem obnavljanja; ključni teorem obnavljanja; prošireni teorem obnavljanja
• Language: english
• URN:NBN: urn:nbn:hr:217:954378
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: viii, 79 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2023-03-01 13:45:50