Stohastička stabilnost Markovljevih lanaca s diskretnim skupom stanja

master's thesis

• Author: Kristina Negovec
• Mentor: Nikola Sandrić (mentor)
• Committee member: Nikola Sandrić (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Na početku ovog rada dajemo uvod u teoriju Markovljevih lanaca na diskretnom skupu stanja. Analiziramo njihovu prolaznost, povratnost, pozitivnu povratnost te nul-povratnost. Glavni rezultat je da ireducibilan Markovljev lanac na konačnom skupu stanja ili pozitivno povratan Markovljev lanac na prebrojivom skupu stanja ima jedinstvenu stacionarnu distribuciju, te da uz dodatni zahtjev aperiodičnosti, prijelazne vjerojatnosti lanca konvergiraju prema stacionarnoj distribuciji. U središnjem dijelu ovoga rada uvodimo martingale te diskutiramo njihova osnovna svojstva. Navodimo teoreme o opcionalnom zaustavljanju te pokazujemo vezu između Markovljevih lanaca i martingala. Nadalje, uvodimo drift operator te razmatramo Foster-Lyapunovljeve kriterije za prolaznost i povratnost. Drift uvjet za povratnost temelji se na postojanju nenegativne, neograničene funkcije \(V\) i odgovarajućeg skupa \(C \in \mathcal{P}(S)\) tako da \(E^x[V(X_1)]-V(x)\le0\), za svaki \(x \in C^c\), dok se drift svojstvo za prolaznost temelji na postojanju nenegativne, ograničene funkcije \(V\) i odgovarajućeg skupa \(C \in \mathcal{P}(S)\) tako da \(E^x[V(X_1)]-V(x)\ge 0\), za svaki \(x \in C^c\). U zadnjem dijelu ovoga rada analiziramo povratnost i prolaznost slučajne šetnje, slučajne šetnje na polupravcu te AR(1) procesa koristeći Lyapunovljevu funkciju te Foster-Lyapunovljeve kriterije za povratnost i prolaznost.

Abstract (english)

In the first part of thesis work we give an introduction to the theory of Markov chains with discrete state space. Further, we discuss their transience, recurrence, positive recurrence and null-recurrence property. The main result is that an irreducible Markov chain with finite state space or positive recurrent Markov chain with countable state space admits a unique stationary distribution and, in addition, by assuming aperiodicity of the chain, transition probabilities converge to the stationary distribution. In the second part of this thesis we recall the notion of martingales and discuss their basic properties. We state optional sampling theorem and discuss the relation between Markov chains and martingales. Further, we introduce a drift operator and discuss Foster-Lyapunov conditions for transience and recurrence. Drift condition for recurrence is based on the existence of nonnegative, unbounded function \(V\) and appropriate set \(C \in \mathcal{P}(S)\) so that \(E^x[V(X_1)]-V(x)\le0\), for every \(x \in C^c\), while drift condition for transience is based on the existence of nonnegative, bounded function \(V\) and the appropriate set \(C \in \mathcal{P}(S)\) so that \(E^x[V(X_1)]-V(x)\ge 0\), for every \(x \in C^c\). In the last part of the thesis, we analyze recurrence and transience of a random walk, random walk on the halfline and AR (1) process using Lyapunov function and Foster-Lyapunov conditions for recurrence and transience criteria.

• Keywords: Markovljevi lanci; martingal; drift operator; Foster-Lyapunovljevi kriteriji za prolaznost i povratnost; slučajna šetnja
• Keywords (english): Markov chains; martingale; drift operator; Foster-Lyapunov conditions for transience and recurrence; random walk
• Language: croatian
• URN:NBN: urn:nbn:hr:217:650657
• 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: 2019-06-07 13:33:16