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Stochastic Equations with Correlated Noise and Their Applications
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Peszat, Szymon (referee) ; Hlubinka, Daniel (referee)
Stochastic Equations with Correlated Noise and Their Applications Ondřej Týbl Doctoral Thesis Abstract Properties of stochastic differential equations with jumps are stud- ied. Lyapunov-type methods are derived to assess long-time behavior of solu- tions and general results are applied in specific cases. In the first case, conditions in terms of the geometric properties of the coefficients for stability in terms of boundedness in probability in the mean are obtained. By means of Krylov Bogolyubov Theorem criterion for existence of invariant measures is given sub- sequentely. In the second case, the long-time behavior refers to existence of an almost sure single-point limit not depending on the initial condition. This result is then applied to get a continuous-time Robbins-Monro type stochastic approximation procedure for finding roots of a given function. 1
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Quantiles for directional data
Fedor, Jakub ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
In this thesis we study a special type of multidimentional data - directional data. The main part of this thesis consists of defining and comparing different ways of order- ing directional data. The most important functions used for ordering directional data presented in this thesis are angular depths. We will describe importnant properties of angular depths and we will discuss wheter each angular depth satisfies the formulated desirable properties. Using previously defined angular depths and median we show two ways of drawing the circular version of a boxplot. 1
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Filtering and prediction of discrete time processes
Šmejkalová, Eva ; Hlubinka, Daniel (advisor) ; Čoupek, Petr (referee)
The thesis focuses on filtering and prediction of discrete time processes. We begin by introducing the elementary notations and theory of discrete-time Markov chains and random walks. We then describe the approach to filtering methods, accompanied by comments, figures and examples. After that we prove one of the fundamental theorems about filtering equations and explain the connection between these equations and the introduction of the chapter by graphically and numerically solving two problems. Finally, we end the paper with a brief description of the topic of prediction and prove a theorem that we then apply to a specific problem. 1
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α-symmetric measures
Ranošová, Hedvika ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
Spherically symmetric measures in Rn are rotationally invariant, indicating that their characteristic functions can be written as a composition of the Euclidean norm with a univariate function. If we replace the Euclidean norm with an ℓα norm, the resulting distributions are known as α-symmetric. This thesis aims to provide a general description of α-symmetric measures and explore various non-trivial examples. The existence of α- symmetric measures for a given α and dimension n ∈ N is discussed, along with the connection between the existence of α-symmetric measures and isometric embedding into Lp spaces through strictly stable distributions. One of the main properties explored in this thesis is the relationship between moments of non-integer order and α-symmetry in distributions. Additionally, several sufficient conditions for the existence and the form of α-symmetric measures are described. In the final chapter, a further generalization of α-symmetric distributions toward quasi-norms is discussed, along with the properties of the resulting concept of pseudo-isotropy. 1
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Limit theorems for dependent random variables
Švarcová, Anna ; Hlubinka, Daniel (advisor) ; Lachout, Petr (referee)
In presented work we will discuss the central limit theorem for dependent random variables. First of all, we brush up the basic version of the theorem and we illustrate it by an example of its use. Then we introduce a definition of the strong mixing condition that allows us to prove the theorem even for dependent random variables. Next, we focus on the assumptions which are essential for the validity of the theorem. The biggest part of the work we deal with its proof. Last of all, we illustrate this theorem with an example which helps us to better understand the main idea of the proof. We simulate this example for specific value of sequences that we define in the wording of the theorem. 1
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Simplicial depth
Mendroš, Erik ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
Depth functions play a crucial role in nonparametric statistics by generalizing orderings, ranks, and quantiles to multivariate data. In our thesis, we provide a comprehensive study of the classical and revised definitions of simplicial depth function, accompanied by detailed and illustrated proofs of some of their proper- ties. Our research also addresses some issues in previous publications and explores potential expansions of those concepts. In the final part of the thesis, we reveal an intriguing connection between simplicial depth and Sylvester's four-point prob- lem, which may have implications for future advancements in this field. 1
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Concentration inequalities for sums
Blatská, Tereza ; Hlubinka, Daniel (advisor) ; Lachout, Petr (referee)
In this bachelor thesis we focus on concentration inequalities for sums of indepen- dent random variables, which are bounded and not necessarily identically distributed. The main pillar of the thesis is Hoeffding's inequality, finding its improvement and other similar inequalities. Inequalities are completed with examples for various probability dis- tributions. In each example there is a theoretical calculation, a simulation for specifically selected parameters and a graphical representation of all the obtained estimates, which was created using the R programming language. 1
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Center-outward ranks and signs and their application in statistical tests
Roubínová, Veronika ; Hudecová, Šárka (advisor) ; Hlubinka, Daniel (referee)
This thesis describes the theory of multivariate rank tests based on center-outward ranks and signs. The definition of the center-outward ranks and signs is based on the measure transportation problem and depends highly on the chosen underlying grid. Sev- eral ways to generate such grids are suggested. Center-outward ranks and signs are then used to construct various test statistics for one-sample testing of location. The main contribution of the work is the introduction of new variants of the one-sample test of location. The proposed test statistics are based on randomized signs and added zero with the usage of the permutation tests for obtaining p-values. The tests are constructed under the assumption of both central or angular symmetry of the underlying distribution. In the end, a simulation study is performed to illustrate the performance of the proposed tests under different settings for several alternatives. 1
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Chebyschev type inequalities
Vachálek, Vladimír ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
Title: Chebyschev type inequalities Author: Vladim'ır Vach'alek Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Abstract: In the presented thesis we deal with Chebyshev type inequalities for bounded random variables. In the first chapter we introduce and prove Hoeffding, Bennett and Bernstein inequalities and explain some relationships. In the second chapter we show how tight are the estimates given by each inequality compared to true probability and to the estimate given by central limit theorem on four distributions with graphical processing of results. Keywords: Chebyshev inequality, Hoeffding inequality, Bennett inequality, Bern- stein inequality 1
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Random walks on the symmetric group - how many times should you shuffle a deck of cards
Hruška, Martin ; Prokešová, Michaela (advisor) ; Hlubinka, Daniel (referee)
This thesis deals with random walks on a symmetric group, namely the models that are used to describe the shuffling of a deck of cards. In this work we focus on the question of mixing speed (the speed of convergence of the marginal distribution of a random walk to its stationary distribution). We ask ourselves a basic question when shuffling cards: how many times do the cards need to be shuffled so that they are already sufficiently randomly distributed. The random walk model, which is a Markov chain, is the mathematical formalization of the card shuffling process. We transfer the card shuffling problem to the problem of estimating the distance between the marginal distribution of this Markov chain and its stationary distribution. We then use standard methods to estimate the convergence rate of the Markov chain to its stationary distribution, such as strong stationary times. 1
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