Galton-Watsonov proces grananja i Kestenovo stablo

master's thesis

• Title (english): Galton-Watson branching process and Chestnut tree
• Author: Luka Jurić
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Matej Mihelčić (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Galton-Watsonov proces ili jednostavan proces grananja je prvi stohastički model razvoja populacije. Proces kreće od jedne jedinke te se svaka jedinka razmnožava jednako i nezavisno od ostalih s određenom distribucijom potomaka. U ovom diplomskom radu bavimo se Galton-Watsonovim stablima. Najprije uvodimo skup diskretnih stabala, definiciju stabla i određene podskupove stabla koji će nam biti od interesa kao što su: širina stabla na određenoj razini, listovi, podstablo stabla iznad određenog čvora i podstablo stabla s određenom visinom. Nakon toga smo spremni definirati Galton-Watsonovo stablo, koje ustvari predstavlja Galton-Watsonov proces kada promatramo širinu stabla na određenoj razini, odnosno broj jedinki iste generacije. S obzirom na distribuciju potomaka, preciznije njenom očekivanju \(m\), razlikujemo kritična (\(m=1\)), sub-kritična (\(m<1\)) i super-kritična (\(m>1\)) Galton-Watsonova stabla. Potvrđujemo da je vjerojatnost izumiranja najmanja fiksna točka funkcije izvodnice distribucije potomaka stabla na intervalu \(\left[0,\:1\right]\), međutim ona je jednaka \(1\) samo za kritična i sub-kritična stabla. Također, super-kritično stablo uvjetno na događaj izumiranja je distribuirano jednako kao sub-kritično stablo s određenom distribucijom potomaka. Na kraju pokazujemo da se na događaju preživljavanja veličina populacije Galton-Watsonovog procesa sa super-kritičnom distribucijom ponaša kao konačna slučajna konstanta pomnožena s \(m^n\), u slučaju kada je \(\mathbb{E}\left[X\ln^+(X)\right]<+\infty\), gdje je \(X\) slučajna varijabla s distribucijom koja je jednaka super-kritičnoj distribuciji potomaka procesa.

Abstract (english)

The Galton-Watson process or the simple branching process is the first stochastic model for population evolution. The process starts from one individual and each individual reproduces equally and independently from the others with a given offspring distribution. In this thesis we deal with Galton-Watson trees. Firstly, we introduce a set of discrete trees, a definition of a tree and certain subsets of a tree that will be of interest to us such as: tree width at a certain level, the set of leaves of a tree, a subtree above a certain node of a tree and a subtree below a certain height of a tree. Subsequently we are ready to define the Galton-Watson tree, which actually represents the Galton-Watson process when we observe the width of the tree at a certain level. In other words, we observe the number of individuals of the same generation. Considering the offspring distribution, more precisely its mean \(m\), we distinguish critical (\(m=1\)), sub-critical (\(m<1\)) and super-critical (\(m>1\)) Galton-Watson trees. We confirm that the probability of extinction is the smallest fixed point of the generating function of the tree’s offspring distribution on the interval \(\left[0,\:1\right]\). However, it is equal to 1 only for critical and sub-critical trees. Also, the super-critical tree conditionally on the extinction event is equally distributed as a sub-critical tree with a special offspring distribution. Finally, we show that at the survival event the population size of the Galton-Watson process with super-critical distribution behaves as a finite random constant multiplied by \(m^n\), when \(\mathbb{E}\left[X\ln^+(X)\right]<+\infty\), where \(X\) is a random variable with a distribution equal to the super-critical offspring distribution of the process.

• Keywords: jednostavan proces grananja; model razvoja populacije; Galton-Watsonova stabla; distribucija potomaka
• Keywords (english): simple branching process; Galton-Watson trees; model for population evolution; offspring distribution
• Language: croatian
• URN:NBN: urn:nbn:hr:217:023497
• Study programme: Title: Finance and Business Mathematics; 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: 2022-10-27 10:42:38