Fluctuation of a simple symmetric random walk

U ovom diplomskom radu proučavamo jednostavne simetrične slučajne šetnje i uočavamo neke naizgled neočekivane rezultate. Kroz primjere uviđamo da neki naizgled neočekivani rezultati imaju veću vjerojatnost nego što smo pretpostavili. Pokazuje se da posjeti ishodištu nisu toliko učestali te da je vjerojatnost da se čestica vratila u ishodište u nekom trenutku jednaka vjerojatnosti da se do tog trenutka uopće nije vratila. Pokazujemo da intiucija zavarava te da prethodni dio šetnje ne utječe na nastavak (kockarova propast). Proučavamo slučajne varijable koje imaju arkus sinusovu distribuciju te pratimo svojstva iste. To nam pokazuje neke zanimljive rezultate koji su iznenađujući. Pokazuje se da ne očekujemo veliki broj promjena vodstva u igri i da postoji velika vjerojatnost da jedna strana provede većinu vremena u vodstvu. Nalazimo dobru aproksimaciju za vjerojatnost u kojoj točki će slučajna šetnja završiti pomoću centralnog graničnog teorema. Preko simulacija i grafova izrađenih simulacija bolje shvaćamo ponašanja jednostavnih simetričnih slučajnih šetnji i uspoređujemo s očekivanim rezultatima. In this paper, we study simple symmetrical random walks and observe some seemingly unexpected results. Through some examples, we see that some seemingly unexpected results have a higher probability than we assumed. It is shown that the visits to the origin are not so frequent and that the probability that the particle has returned to the origin at some point in time is equal to the probability that it has not returned at all until that moment. We show that intuition is deceiving and that the previous part of the walk does not a ect the continuation (gambler’s ruin). We study random variables that have an arcsine distribution and monitor its properties. This shows us some interesting results that are surprising. It turns out that we don’t expect a lot of lead changes in the game and that there is a big chance one side spends most of the time in the lead. An interesting result is that if we fix that our walk ends at the origin, the distribution of spending time in the positive quadrant ceases to be an arcsine distribution and becomes unfornm. We find a good approximation for the probability at which point the random walk will end using the central limit theorem. Using simulations and graphs of the created simulations, we better understand the behavior of simple symmetric random walks and compare them with the expected results.