# Zakoni arkus sinusa

Prša, Marija (2014) Zakoni arkus sinusa. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

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.