Markovljevi lanci s atomima

master's thesis

• Title (english): Markov chains with atoms
• Author: Jakov Šikić
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj se rad bavi Markovljevim lancima na općenitom skupu stanja. Radi jednostavnosti, bavimo se Markovljevim lancima s atomima. Takvi lanci predstavljaju važnu klasu slučajnih procesa te su primjenjivi na široki spektar nematematičkih područja. U prvom se poglavlju iznosi sažetak osnovnih rezultata teorije Markovljevih lanaca na diskretnom skupu stanja. Drugo poglavlje je iskorišteno za uvođenje pojma Markovljeve jezgre te definicije Markovljevih lanaca i homogenih Markovljevih lanaca na općenitom skupu stanja. Teoremom 2.1.11 dokazana je važna karakterizacija uvjeta homogenih Markovljevih lanaca. U nastavku se poglavlja dokazuje jako Markovljevo svojstvo te iskazuju neke njegove posljedice koje će biti korištene u nastavku. Na početku trećega poglavlja, definicijom 3.1.1, uvodi se pojam atoma. Nadalje, analogno diskretnom slučaju, definiraju se i dokazuju karakterizacije povratnosti i prolaznosti atoma. Teoremom 3.2.2 dokazuju se nužni i dovoljni uvjeti za postojanje inavarijantne mjere i distribucije. Nadalje, u korolaru 3.3.3 dokazuje se ergodski teorem. Krunu razmatranja predstavlja centralni granični teorem za homogene Markovljeve lance, dokazan teoremom 3.4.2. U završnom se poglavlju rada prikazuju neki primjeri Markovljevih lanaca na općenitom skupu stanja, pri čemu je najviše pozornosti dano Gl/G/1 repu.

Abstract (english)

This work deals with Markov chains on a general state space. For simplicity, we deal with atomic Markov chains. Such chains represent an important class of stochastic processes and are applicable to a wide range of non-mathematical fields. In the first chapter we survey basic results of Markov chain theory on a discrete state space. The second chapter is dedicated to introduction of a Markov kernel, also the definition of Markov chains and homogeneous Markov chains on a general state space. Theorem 2.1.11 proves an important characterization of homogeneous Markov chains. Moreover, we prove the strong Markov property and state some of its consequences which will be used in later chapters. In the beginning of the third chapter, through definition 3.1.1, we introduce the concept of an atom. Furthermore, in analogy with a discrete state space we define and prove the characterization of atomic recurrence and transience. Theorem 3.2.2 proves necessary and sufficient conditions for the existence of an invariant measure and distribution. Corollary 3.3.3 proves the ergodic theorem. In the end through theorem 3.4.2 we prove the central limit theorem for homogeneous Markov chains. In the last chapter we present some examples of Markov chains on a general state space, where the most notable example is a Gl/G/1 queue.

• Keywords: teorija Markovljevih lanaca; Markovljeva jezgra; ergodski teorem; Gl/G/1 rep
• Keywords (english): Markov chain theory; Markov kernel; ergodic theorem; Gl/G/1 queue
• Language: croatian
• URN:NBN: urn:nbn:hr:217:932818
• 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: 2021-11-08 11:37:38