Zakoni arkus sinusa

master's thesis

• Author: Marija Prša
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U prvom dijelu rada bavili smo se slučajnim šetnjama. Inicijalno smo promatrali jednostavnu simetričnu slučajnu šetnju duljine \(2n\) koja kreće iz ishodišta i dokazali nekoliko poznatih zakona arkus sinusa. Posebno, slučajna varijabla koja označava trenutak zadnjeg posjeta nuli u slučajnoj šetnji, slučajna varijabla koja označava broj segmenata \((k-1, S_{k-1}) \to (k, S_k)\) koji leže iznad \(x\)-osi, slučajna varijabla koja za danu slučajnu šetnju duljine \(2n\) daje najmanji \(j\) indeks takav da je \(S_j = S_{2n}\) te slučajna varijabla koja označava indeks prvog postizanja maksimuma imaju diskretnu arkus sinus distribuciju reda \(n\). Pomoću teorema Sparre-Andersena poopćili smo zakone arkus sinusa i na općenitije slučajne šetnje. Za slučajne šetnje za koje vrijedi da je distribucija koraka slučajne šetnje simetrična i \(\mathbb{P}(S_m = 0) = 0\), za svaki \(m \geqslant 1\) vrijedi da slučajna varijabla koja označava broj točaka iznad \(x-\)osi ima diskretnu arkus sinus distribuciju. Također, slučajne varijable koje označavaju trenutak prvog maksimuma te zadnjeg minimuma imaju diskretnu arkus sinus distribuciju. Poseban slučaj općenite slučajne šetnje koja zadovoljava navedena svojstva je slučajna šetnja sa simetričnom i neprekidnom funkcijom distribucije koraka. U drugom dijelu rada promatrali smo Brownovo gibanje na intervalu [0, 1]. Pomoću Markovljevog svojstva Brownovog gibanja i principa refleksije dokazali smo da slučajna varijabla koja označava zadnji posjet nuli Brownovog gibanja prije trenutka 1 ima arkus sinus distribuciju. Nakon što smo vidjeli da Brownovo gibanje poprima jedinstveni maksimum na intervalu [0, 1] gotovo sigurno, dokazali smo da i slučajna varijabla koja označava trenutak postizanja maksimuma ima arkus sinus distribuciju. Na kraju smo pokazali dokaz i poznatog zakona arkus sinusa poznatog pod nazivom Levyev zakon arkus sinusa. On se odnosi na slučajnu varijablu koja označava vrijeme koje Brownovo gibanje provede iznad \(x-\)osi. U dokazu je korišten Donskerov princip invarijantnosti, koji kaže da skalirana slučajna šetnja konvergira po distribuciji standardnom Brownovom gibanju, i prethodno dokazani zakoni za maksimum Brownovog gibanja te jednostavne slučajne šetnje.

Abstract (english)

In the first part of the thesis we considered random walks. We studied a simple symmetric random walk with \(2n\) steps which starts from the origin and proved several well-known arcsine laws. In particular, random variable denoting the time of the last zero before \(2n\), random variable denoting the number of segments \((k-1, S_{k-1}) \to (k, S_k)\) that lie in the upper half plane, random variable denoting the minimal index \(j\) such that \(S_j = S_{2n}\) and random variable denoting the index of the first maximum have discrete arcsine distribution of order \(n\). Using Sparre-Andersen theorem we showed how one can extend the arcsine laws to very general random walks. For random walks with symmetric step distribution and with the property that for every \(m \geqslant 1\) is \(\mathbb{P}(S_m = 0) = 0\), a random variable that indicates the number of points in the upper half plane has discrete arcsine distribution. Also, the random variables denoting the moment of the first maximum and last minimum have a discrete arcsine distribution. A special case of general random walk satisfying the above properties is random walk with symmetric and continuous step distribution. In the second part we studied the Brownian motion on the interval [0, 1]. Using Markov property of Brownian motion and the reflection principle, we proved that random variable that denotes the last zero of Brownian motion in [0, 1] is arcsine distributed. Once we showed that for Brownian motion on [0, 1] the global maximum is attained at a unique time we have proved that the random variable that denotes the time at which a Brownian motion achieves its maximum is also arcsine distributed. In the end we showed proof of famous arcsine law known as Levy’s arcsine law. It refers to the random variable that indicates the time that Brownian motion spends in the upper half plane. In the proof we used Donsker’s invariance principle, which says that scaled random walk converges in distribution to a standard Brownian motion, as well as the already established laws for the maximum of Brownian motion and simple random walks.

• Keywords: zakoni arkus sinusa; slučajne šetnje; Sparre-Andersenov teorem; Brownovo gibanje; Levyev zakon arkus sinusa; Donskerov princip invarijantnosti
• Keywords (english): arcsine laws; random walks; Sparre-Andersen theorem; Brownian motion; Levy’s arcsine law; Donsker’s invariance principle
• Language: croatian
• URN:NBN: urn:nbn:hr:217:067396
• Study programme: Title: Finance and Business Mathematics; 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: 2019-02-01 10:16:23