Modeli urni i martingalne metode

U ovom radu bavimo se modelima urni. Modeli urni su jako poznati i vrlo stari modeli pomoću kojih se može opisati čitav niz problema iz primijenjene vjerojatnosti. Najvažnije pitanje koje se javlja prilikom proučavanja modela urni je pitanje asimptotskog ponašanja distribucije broja kuglica pojedine boje u urni. Postoji mnogo različitih pristupa pronalasku odgovora na to pitanje. Mi u ovom radu opisujemo martingalni pristup tom problemu. Na početku rada donosimo kratak uvod u teoriju martingala, a zatim opisujemo Bagchi-Pal model urne na kojem smo i ilustrirali martingalni pristup rješavanju problema pronalaska asimptotske distribucije broja kuglica pojedine boje u urni. Osim opisa Bagchi-Pal modela urne, donosimo i vrlo kratak opis dva najpoznatija modela urni u literaturi, a to su shema Georgea Pólye i shema Bernarda Friedmana. Navodimo i najvažnije rezultate vezane uz ta dva modela te komentiramo kako se oni uklapaju u rezultate koje donosimo kasnije, a koji se ticu Bagchi-Pal modela urni. Prvi veliki rezultat koji donosimo je takozvani zakon velikih brojeva za urne koji govori o tome kako se nakon dugo vremena, u slučaju Bagchi-Pal modela urne koji zadovoljava još neke dodatne uvjete, ponaša omjer broja kuglica pojedine boje u urni i broja koraka. Drugi veliki rezultat koji donosimo je centralni granični teorem za urne. I taj rezultat se odnosi na Bagchi-Pal model urne koji zadovoljava neke dodatne uvjete. Ovaj rezultat nam govori o distribuciji udjela kuglica pojedine boje u urni nakon dugo vremena. Kako bismo ilustrirali gore spomenuta dva rezultata, na kraju rada donosimo simulacije izvlačenja kuglica iz urne i ubacivanja novih kuglica u urnu koje smo napravili pomoću programskog jezika R. Grafovi čije slike smo prikazali u zadnjem poglavlju jako lijepo ilustriraju dobivene rezultate.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