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Existence and uniqueness of the distribution of a random measure given by finite dimensional projections
Jurčo, Adam ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
Title: Existence and uniqueness of the distribution of a random measure given by finite dimensional projections Author: Adam Jurčo Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jan Rataj, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis deals with the existence and uniqueness of the distribu- tion of a random measure given a system of finite-dimensional distributions. A random measure can be interpreted as a particular system of random variables. Conversely, we will want to know what conditions would allow a system of random variables to be extended to a random measure and if this extension is unique. We will start with a consistent system of finite-dimensional distributions and use Daniell-Kolmogorov theorem to find the necessary and sufficient conditions for the existence of such extension. A counterexample will be included to show that it is not possible to use this theory for random signed measures. Keywords: Random measure, point process, finite-dimensional distributions. 1
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Random closed sets and particle processes
Stroganov, Vladimír ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
In this thesis we are concerned with representation of random closed sets in Rd with values concentrated on a space UX of locally finite unions of sets from a given class X ⊂ F. We examine existence of their repre- sentations with particle processes on the same space X, which keep invariance to rigid motions, which the initial random set was invariant to. We discuss existence of such representations for selected practically applicable spaces X: we go through the known results for convex sets and introduce new proofs for cases of sets with positive reach and for smooth k-dimensional submanifolds. Beside that we present series of general results related to representation of random UX sets. 1
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Density of Minkowski functionals of stationary random sets
Dohnálek, Filip ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Density of Minkowski functionals of stationary random sets Author: Bc. Filip Dohnálek Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jan Rataj, CSc., Mathematical Institute of Charles University Abstract: In the presented work we can find the created theory of random closed excursion set generated by means of Gaussian real random field. We specialize in a real random field, which is defined on the regular stratified manifold. The text includes a determination of conditions for a random field and stratified manifold in which densities of the intrinsic volumes for excursion sets exist. Then subsequently attributes and relations of the excursion set are derived for the existence of densities of the intrinsic volumes. Finally, a simulated study is made where we compare theoretical and estimated values of densities. This is followed by a discussion on the results, which we compare to the Boolean model. Keywords: Densities of the intrinsic volume, Excursion set, Manifold, Real random field
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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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Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor) ; Rataj, Jan (referee)
Title: Nonabsolutely convergent integrals Author: Kristýna Kuncová Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jan Malý, DrSc., Department of Mathematical Analysis Abstract: Our aim is to introduce an integral on a measure metric space, which will be nonabsolutely convergent but including the Lebesgue integral. We start with spaces of continuous and Lipschitz functions, spaces of Radon measures and their dual and predual spaces. We build up the so-called uniformly controlled integral (UC-integral) of a function with respect to a distribution. Then we investigate the relationship between the UC-integral with respect to a measure and the Lebesgue integral. Then we introduce another kind of integral, called UCN-integral, based on neglecting of small sets with respect to a Hausdorff measure. Hereafter, we focus on the concept of n-dimensional metric currents. We build the UC-integral with respect to a current and then we proceed to a very general version of Gauss-Green Theorem, which includes the Stokes Theorem on manifolds as a special case. Keywords: Nonabsolutely convergent integrals, Multidimensional integrals, Gauss-Green Theorem 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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Nonstationary particle processes
Jirsák, Čeněk ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Nonstacionary particle processes Author: Čeněk Jirsák Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Rataj, CSc., Mathematical Institute, Charles University Supervisor's e-mail address: rataj@karlin.mff.cuni.cz Abstract: Many real phenomena can be modeled as random closed sets of different Hausdorff dimension in Rd . One of the main characteristics of such random set is its expected Hausdorff measure. In case that this measure has a density, the density is called intensity function. In present paper we define a nonparametric kernel estimation of the intensity function. The concept of Hk -rectifiable set has a key role here. Properties of kernel estimation such as unbiasness or convergence behavior are studied. As the esti- mation may be difficult to compute precisely numerical approximations are derived for practical use. Parametric models are also briefly mentioned and the kernel estimation is used with the minimum contrast method to estimate the parameters of the model. At last the suggested methods are tested on simulated data. Keywords: stochastic geometry, intensity measure, random closed set, kernel estimation 1
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Set-indexed stochastic processes
Schenk, Martin ; Rataj, Jan (referee) ; Pawlas, Zbyněk (advisor)
This thesis deals with the problem of estimating the joint probability distribution of a marked process' parameters from a censored data. First, a Nelson-Aalen estimator of the cumulative hazard rate for one-dimensional case is constructed. This estimator is then smoothed by using a kernel function estimator. Then, a Kaplan-Meier estimator of the survival function is brought in. Further, a theory of set-indexed random processes is built up to be a base for the construction of a generalized Nelson-Aalen estimator of the cumulative hazard rate, which is then again smoothed. For a special case, a generalized Kaplan-Meier estimator of the multidimensional survival function is constructed. The application of the mentioned generalized estimators is shown on a particular case. These estimators are then used on simulated data.
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