Markovljevi lanci na općenitom skupu stanja i povratnost

Lisičar, Arijan (2014) Markovljevi lanci na općenitom skupu stanja i povratnost. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

In the first chapter we list the main terms in the theory of Markov chains on discrete state space. After that, in the second chapter, we expand that theory to Markov chains on general state space. In Definition 2.1.1 we introduce terms like the kernel of a Markov chain as well as occupation times η, first return times τ, first hitting times σ and stopping times ζ. We show that Markov chains on general state space cam have similar properties as on discrete state space. We define uniform and geometric ergodicity and show, in Theorems 2.5.7 and 2.5.13, the conditions needed for our chain to have those properties. In chapter three we focus on a smaller class of Markov chains, so-called Harris chains. Function Q(x, A) is the probability of chain X visiting set A infinite number of times, considering he starts from state x. On the other hand, function L(x, A) is the probability of chain X to return to set A. In Theorem 3.2.2, we show how the properties of the function Q can be linked to those of the function L. Finally, in Theorems 3.4.2 and 3.4.3 we show the sufficient conditions Harris chains need to be ergodic. We finish this thesis with a couple of examples of Markov chains on general state space: the autoregressive process of order k and the well known storage model.

Item Type: Thesis (Diploma thesis)
Supervisor: Basrak, Bojan
Date: 2014
Number of Pages: 55
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 03 Jun 2015 12:19
Last Modified: 03 Jun 2015 12:19
URI: http://digre.pmf.unizg.hr/id/eprint/3996

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