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master's thesis
Markovljevi lanci unatrag
Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. urn:nbn:hr:126:238895
Bencetić, Doris
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Pure Mathematics and Mathematics Teaching
Algebra, Analysis and Number Theory Research Group

Institutional repository: Repository of School of Applied Mathematics and Informatics

Cite this document

Bencetić, D. (2020). Markovljevi lanci unatrag (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:238895

Bencetić, Doris. "Markovljevi lanci unatrag." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:238895

Bencetić, Doris. "Markovljevi lanci unatrag." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:238895

Bencetić, D. (2020). 'Markovljevi lanci unatrag', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 19 January 2025, https://urn.nsk.hr/urn:nbn:hr:126:238895

Bencetić D. Markovljevi lanci unatrag [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2020 [cited 2025 January 19] Available at: https://urn.nsk.hr/urn:nbn:hr:126:238895

D. Bencetić, "Markovljevi lanci unatrag", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:126:238895

Metadata
TitleMarkovljevi lanci unatrag
Title (english)Markov Chains Backwards
AuthorDoris Bencetić
MentorMirta Benšić (mentor)
Committee memberDragana Jankov Maširević (predsjednik povjerenstva)
Committee memberMirta Benšić (član povjerenstva)
Committee memberDanijel Grahovac (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Pure Mathematics and Mathematics Teaching)
(Algebra, Analysis and Number Theory Research Group)
Osijek
Defense date and country2020-10-20, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Other Mathematical Disciplines
Abstract
U ovom diplomskom radu upoznali smo se s Markovljevim lancima unatrag u diskretnom vremenu. U prvom poglavlju definirali smo Markovljeve lance u diskretnom vremenu.Također, uveli smo važne pojmove kao što su matrica prijelaznih vjerojatnosti, vrijeme prvog posjeta, vrijeme prvog povratka, invarijantna mjera, graf stanja itd. Klasificirali smo stanja Markovljeva lanca te naveli važne rezultate za razumijevanje Markovljevih lanaca. U drugom poglavlju, koje je i glavni dio rada, definirali smo Markovljeve lance unatrag te dokazali njihova najvažnija svojstva. Naveli smo jednadžbu ravnoteže za matrice P, Pˆ i mjeru λ a kasnije i složeniju verziju jednadžbe ravnoteže te jednadžbu ravnoteže za matrice n-koračnih prijelaznih vjerojatnosti. Osim toga, naveli smo i jedan jednostavan primjer Markovljeva lanca unatrag. Od posebnog interesa bili su nam Markovljevi lanci unatrag čije se matrice prijelaznih vjerojatnosti podudaraju s originalnim matricama prijelaznih vjerojatnosti. Oni su poznati pod imenom reverzibilni Markovljevi lanci. Dokazali smo njihova bitna svojstva. Nakon toga, u zasebnim potpoglavljima raspravili smo reverzibilnost Ehrenfestovog lanca, reverzibilnost Bernoulli-Laplaceovog lanca i reverzibilnost lanaca radanja i umiranja. Diskutirali smo i slučajne šetnje na grafovima u terminima reverzibilnih lanaca te lance pouzdanosti u terminima Markovljevih lanaca unatrag.
Abstract (english)
In this thesis, we introduce Markov chains backwards in discrete time. In the first chapter, we defined Markov chains in discrete time. Also, we have defined important terms such as transition probability matrix, time of the first visit, time of the first return, invariant measure, state graph etc. We classified Markov chain states and presented results that are important for understanding Markov chains. In the second chapter, which is the main part of this thesis, we have defined Markov chains backwards and we have proved their properties. We introduced balance equation which holds for matrices P, Pˆ and for measure λ. Later we introduced more complex balance equation so as balance equation that holds for n state transition probability matrices. Further, we set an example of Markov chain backwards. Markov chains backwards whose transition probability matrices are identical to original transition probability matrices were of special interest to us. They are known as reversible Markov chains. We also proved some important properties of reversible Markov chains. Furthermore, we discussed reversibility of Ehrenfest chain, reversibility of Bernoulli-Laplace chain and reversibility of birth and death chains in separate subsections. Also, we discussed random walks on graphs in terms of reversible Markov chains and birth- death chains in terms of Markov chains backwards
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:238895
Study programmeTitle: Mathematics; specializations in: Financial and Statistical Mathematics, Mathematics and Computer Science, Industrial and Applied Mathematics
Course: Financial and Statistical Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2020-10-30 11:19:44