Granični teoremi za Galton-Watsonove procese

master's thesis

• Title (english): Boundary theorems for Galton-Watson processes
• Author: Dario Domjanić
• Mentor: Zoran Vondraček (mentor)
• Committee member: Zoran Vondraček (predsjednik povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatramo granično ponašanje Galton-Watsonovih procesa. Ti procesi opisuju razvoj populacije koja kreće od jedne jedinke. Svaka jedinka se razmnožava nezavisno od ostalih i ima jednaku distribuciju potomaka. Pokazujemo da ovakvi procesi imaju svojevrsnu dualnost: populacija ili izumre ili sve više raste. Ovisno o očekivanom broju potomaka m, Galton-Watsonove procese dijelimo na subkritične (m < 1), kritične (m = 1) i superkritične (m > 1). Subkritični i kritični procesi izumru g.s., dok superkritični preživljavaju s pozitivnom vjerojatnošću. Postavlja se pitanje koliko brzo subkritični i kritični procesi izumiru te kolika je brzina rasta superkritičnih procesa u slučaju preživljavanja. Pokazujemo da je kod subkritičnih i superkritičnih procesa glavno pitanje kada je mn prava brzina izumiranja, odnosno rasta. U oba slučaja ispada da je to ekvivalentno ”L log L uvjetu” koji glasi \(\mathbb{E} [L\log^+ L]<\infty\), gdje je L slučajna varijabla koja predstavlja broj djece neke jedinke u sljedećoj generaciji. Ukoliko taj uvjet nije zadovoljen, subkritični procesi izumiru još brže od \(m^n\), a kritični procesi rastu sporije od \(m^n\). Štoviše, u tom slučaju rast superkritičnih procesa nije puno sporiji. Kod kritičnih procesa imamo malo drugačiju situaciju. Ovdje je brzina izumiranja dana s \(1/n\) ako je \(Var(L)<\infty\), a brzina je još veća ako je \(Var(L)=\infty\).

Abstract (english)

In this thesis we have studied the limit behavior of Galton–Watson processes. Such processes describe the evolution of the population which starts with one individual. All individuals reproduce independently from each other and have the same offspring distribution. We show that these processes have a certain dichotomy: they either become extinct or population size explodes. Depending on mean of the offspring distribution m, Galton–Watson process is called subcritical if m < 1, critical if m = 1, and supercritical if m > 1. Subcritical and critical processes become extinct a.e., while supercritical processes survive with positive probability. Finer question is how fast subcritical and critical processes become extinct and what is the growth rate of supercritical processes if they survive. We show that the main question concerning subcritical and supercritical processes is when does \(m^n\) give the right decay rate and the right growth rate respectively. In both cases, it turns out that this is equivalent to ”L log L” criterion which states \(\mathbb{E} [L\log^+ L]<\infty\), where L is a random variable denoting the number of children of some individual. If this condition isn’t true, subcritical processes die out faster than \(m^n\), while the growth rate of supercritical processes is less than \(m^n\). Moreover, in that instance supercritical processes grow just slightly slower than \(m^n\). The behavior of critical processes is somewhat different. They die out with the rate of \(1/n\) if \(Var(L)<\infty\), whereas the pace of extinction is even greater if \(Var(L)=\infty\).

• Keywords: razvoj populacije; distribucija potomaka; subkritični procesi; kritični procesi; superkritični procesi
• Keywords (english): population development; distribution of offspring; subcritical processes; critical processes; supercritical processes
• Language: croatian
• URN:NBN: urn:nbn:hr:217:591028
• 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: 2021-12-14 12:20:29