Modeli urni i martingalne metode

Šebek, Stjepan (2014) Modeli urni i martingalne metode. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

In this paper we are dealing with urn models. Urn models are well known and very old models that can help us describe a lot of different problems from applied probability. The most important question that arises when we study urn models is the question about asymptotic behaviour of distribution of balls of different colours in our urn. There are many different ways to address this question. In our paper, we present martingale approach to this problem. At the beginning of this paper, we present short introduction into martingale theory, and after that we describe Bagchi-Pal urn model which we used to illustrate martingale approach to finding asymptotic distribution of the number of balls of different colours in an urn. Beside description of Bagchi-Pal urn model, we gave short description of two most famous urn models in literature. Those are Pólya urn model and scheme of Bernard ´ Friedman. We quote the most important results concerning that two models and after that we comment how that results fit into results that we present later and that are concerning Bagchi-Pal urn models. First important result that we proved in this paper is so called law of large numbers for urns. That result tells us how ratio of the number of balls of some colour and the number of steps is behaving after long period of time in Bagchi-Pal urn model that satisfies some additional constraints. Second big result that we proved in our paper is central limit theorem for urns. Assumption of that theorem is that we have Bagchi-Pal urn model that satisfies some additional constraints, just like in the law of large numbers for urns. That result tells us about asymptotic behaviour of distribution of proportion of balls of some colour in an urn after long period of time. In order to illustrate the above mentioned results, at the end of our paper we made simulations of drawing balls from urn and adding new balls to the urn. For simulating that process we used programming language R. We included some pictures in our paper that are very nice illustration of proved results.

Item Type: Thesis (Diploma thesis)
Supervisor: Vondraček, Zoran
Date: 2014
Number of Pages: 56
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 07 Oct 2015 11:02
Last Modified: 07 Oct 2015 11:02
URI: http://digre.pmf.unizg.hr/id/eprint/4256

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