Markovljevi lanci s atomom

master's thesis

• Title (english): Markov chains with an atom
• Author: Ivan Protrka
• Mentor: Nikola Sandrić (mentor)
• Committee member: Nikola Sandrić (predsjednik povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U prvom poglavlju promatramo Markovljeve lance s diskretnim vremenom i diskretnim skupom stanja. Prisjećamo se pojmova kao što su vremena zaustavljanja, povratnost, prolaznost, aperiodičnost itd. U drugome poglavlju i dalje promatramo vremenski diskretne Markovljeve lance, ali na općenitom skupu stanja s naglaskom na Markovljeve lance s atomom. Započinjemo s definicijom Markovljevog lanca na općenitom skupu stanja te uvodimo pojam prijelazne jezgre. Zatim definiramo vremena zaustavljanja te pokazujemo da mnogi važni rezultati, poput npr. jakog Markovljevog svojstva, vrijede i na općenitom skupu stanja. Zatim uvodimo pojam tzv. ψ-ireducibilnosti nakon čega definiramo središnji pojam ovog rada - atom. Nadalje uvodimo pojmove prolaznosti, povratnosti i aperiodičnosti, pokazujemo da se kod lanaca s atomom ta svojstva definiraju i provjeravaju slično kao u diskretnom slučaju te navedeno pokazujemo na nekim jednostavnijim primjerima kao što je slučajna šetnja na polupravcu. Zatim uvodimo pojmove invarijantne mjere i pozitivne povratnosti te u Kacovom teoremu pokazujemo da su kod lanaca s atomom nužni i dovoljni uvjeti za postojanje invarijantne vjerojatnosne mjere isti kao i u diskretnom slučaju. U zadnjem dijelu definiramo ergodične atome i ergodične Markovljeve lance te pokazujemo pod kojim uvjetima vrijede navedena svojstva.

Abstract (english)

In the first chapter we consider discrete-time Markov chains on countable state space. We ldiscuss several properties like recurrence, transience, periodicity etc. In the second chapter we consider discrete-time Markov chains, but state space is now general. We start from definitions of Markov chains on general state space and transition probability kernel. After that we define stopping times and show that many important results, such as strong Markov property, are also valid on a general state space. Then we introduce the notion of ψ-irreducibility and define the central notion of this work - the atom. Furthermore, we introduce the concepts of transience, recurrence and aperiodicity. We show that, if we have an atomic chain, these properties are defined and checked similarly as on discrete state space. We give a few simple examples like the random walk on a half line. Then we introduce the concepts of invariant measure and positive recurrence for atomic chains and in Kac’s Theorem we show that the necessary and sufficient conditions for the existence of an invariant probability measure are the same as in the discrete state space. In the last part we define ergodicity and \(f \)-ergodicity and consider the conditions needed for chain with atom to have those properties.

• Keywords: vremenski diskretni Markovljevi lanci; jako Markovljevo svojstvo; Kacov teorem; ergodični Markovljevi lanci
• Keywords (english): discrete-time Markov chains; strong Markov property; Kac’s Theorem; ergodicity
• Language: croatian
• URN:NBN: urn:nbn:hr:217:419547
• 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: 2021-11-05 12:29:26