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Segment point processes
Honzl, Ondřej ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
Naz-ev prace: Bodove proeesy usecek Aut.or: Ondrej Ilonzl Katedra: Katedra. pravdepodobnosti a matematicke statist,iky Vodonci bakalafske praee: RNDr. Zbynek Pawlas, Ph.D. e-mail vedouciho: zbynek.pawlas'O'mfl'.cmn.cz Abstrakt: Prace obsahnje strucny uvod do teoric bodovych procesii na nplnem se- parabilnim lokalne koutpakt iiiiu metrickem prost.oru. Hamcove je ziuinen si>ccii'ilni I>fipad st,acionariiilio ])roc:csu kouipaktiiich ninozin. Dale sc jiran1 vice1 /aincfujo na, Poissonuv prort\ usccek sc /nainyni ro/dcMrnim typickrho /rnu. V roviniirin pfi])aclo pak ukazujc n'i/nr odliady dolkovr int.i'ii/ily I'oissonova jiroccsu usccck, kU're jsou drfinovany na /aklade udaju /iskanych v okuc poxorovani. Hlavnim zajinoin prace st1 stava porovnavani tcclito odhadu die jujich rozpt.ylu. Cilem to- hoto sroviiavani ina byi stanovoni niezc \vlikoHti okna, klcra. fika, dokud jc Icpsi pou/it slozitojsi odhad a odkdy je naopak ro/ninno pouzit odliad. jclioz vvpocot jo snazsi, ale kl.cry pft'ilpoklada., zc inatiie vice iniormaci u po/orovanein JJI'OCCHU. Klicova slova: I'oissonuv i>roces. hodovy proces usccek, odhad delkove intonzit.y Title: Segment point Autlior: Ondrej Ilonzl Deijartuient: DepartiiieiiL ol'Prol)a.l>ilit.y and Mathcinalical Statistics Supervisor: KNDr. Zbyuek Pawlas, Ph.I). Supervisor's (^niail address:...
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Random closed sets
Stroganov, Vladimír ; Honzl, Ondřej (advisor) ; Rataj, Jan (referee)
In this bachelor thesis we are concerned with basic knowledge in random set theory. We define here such terms, as capacity functional, se- lection, measurable and integrable multifunction, Castaing representation and Aumann expectation of random closed set. We present Choquet theo- rem, Himmelberg measurability theorem, theorems of properties of selections and expectation. We present also several examples which illustrate the the- ory. 1
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On Selected Geometric Properties of Brownian Motion Paths
Honzl, Ondřej
Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,...
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On Selected Geometric Properties of Brownian Motion Paths
Honzl, Ondřej
Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,...
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On Selected Geometric Properties of Brownian Motion Paths
Honzl, Ondřej ; Rataj, Jan (advisor) ; Beneš, Viktor (referee) ; Mrkvička, Tomáš (referee)
Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,...
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Coupling and the speed of convergence of Markov chains
Macháček, Adam ; Prokešová, Michaela (advisor) ; Honzl, Ondřej (referee)
In the present work we study two methods for estimating the rate of convergence of marginal distribution of a discrete-time irre- ducible and aperiodic Markov chain to its stationary distribution (i.e. the speed of mixing). Firstly, we concentrate on deriving an upper bound for the rate of convergence by using the coupling technique. Moreover, we will also define the total variation distance and explain its role in the es- timation of the speed of mixing. In the second part we will study the me- thod of strong uniform times. Both methods are described in detail and several basic theorems are proved. Finally we demonstrate the use of both approaches on a number of models, particularly on the random walk on a hypercube and on shuffling a deck of cards. 1
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Random closed sets
Stroganov, Vladimír ; Honzl, Ondřej (advisor) ; Rataj, Jan (referee)
In this bachelor thesis we are concerned with basic knowledge in random set theory. We define here such terms, as capacity functional, se- lection, measurable and integrable multifunction, Castaing representation and Aumann expectation of random closed set. We present Choquet theo- rem, Himmelberg measurability theorem, theorems of properties of selections and expectation. We present also several examples which illustrate the the- ory. 1
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