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Beveridge-Nelson decomposition and its applications
Masák, Štěpán ; Prášková, Zuzana (advisor) ; Lachout, Petr (referee)
In this work we deal with the Beveridge-Nelson decomposition of a linear process into a trend and a cyclical component. First, we generalize the decom- position for multidimensional linear process and then we use it to prove some of the limit theorems for the process and its special cases, processes VAR and VARMA. Further, we define the concept of cointegration and introduce the po- pular VEC model for cointegrated time series. Finally, we show a method how to deal with infinite sums appearing in calculation of the Beveridge-Nelson decom- position and apply it to real data. Then we compare the results of this method with approximations using partial sums.
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Coupon's collector problem
Nývltová, Veronika ; Pawlas, Zbyněk (advisor) ; Bártek, Jan (referee)
In the presented work we are looking for the answer to the question how many purchases must be made for obtaining a collection of cards. As a collection we understand all types of cards, which are packaged with products or we consider a collection of chosen types of these cards. First it is assumed that all cards are uniformly distributed. The number of required purchases is random and we derive its mean value, variance and probability distribution. We study limit behaviour when the number of types of cards is going to infinity. We are looking for the answer to the same question in the case of collecting several collections of cards at the same time. These collections could be complete or incomplete. In the case that cards are not uniformly distributed we describe mean value and variance of the number of purchases necessary for acquiring several collections of cards.
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Proofs of the strong law of large numbers
Odintsov, Kirill ; Štěpán, Josef (advisor) ; Staněk, Jakub (referee)
This thesis concentrates on the Strong Law of Large Numbers. It features two proofs of this law. The first is less general, but simpler Borel's proof. The second one is more complex. It uses Kronecker's lemma and Kolmogorov-Khinchin's theorem, which is proven by Kolmogorov's inequality. The text includes all the necessary auxiliary theorems and lemmas along with their proofs. Since all the proofs are explored in a great detail this text is suitable for readers with only basic knowledge of probability theory and measure theory. Furthermore it contains numerous practical and mathematical examples thought out the whole text. Finally to demonstrate the importance of Strong Law of Large Numbers the text features four important applications of the law in mathematics.
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Beveridge-Nelson decomposition and its applications
Masák, Štěpán ; Prášková, Zuzana (advisor) ; Lachout, Petr (referee)
In this work we deal with the Beveridge-Nelson decomposition of a linear process into a trend and a cyclical component. First, we generalize the decom- position for multidimensional linear process and then we use it to prove some of the limit theorems for the process and its special cases, processes VAR and VARMA. Further, we define the concept of cointegration and introduce the po- pular VEC model for cointegrated time series. Finally, we show a method how to deal with infinite sums appearing in calculation of the Beveridge-Nelson decom- position and apply it to real data. Then we compare the results of this method with approximations using partial sums.
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|
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Coupon's collector problem
Nývltová, Veronika ; Pawlas, Zbyněk (advisor) ; Bártek, Jan (referee)
In the presented work we are looking for the answer to the question how many purchases must be made for obtaining a collection of cards. As a collection we understand all types of cards, which are packaged with products or we consider a collection of chosen types of these cards. First it is assumed that all cards are uniformly distributed. The number of required purchases is random and we derive its mean value, variance and probability distribution. We study limit behaviour when the number of types of cards is going to infinity. We are looking for the answer to the same question in the case of collecting several collections of cards at the same time. These collections could be complete or incomplete. In the case that cards are not uniformly distributed we describe mean value and variance of the number of purchases necessary for acquiring several collections of cards.
|
|
|
Proofs of the strong law of large numbers
Odintsov, Kirill ; Štěpán, Josef (advisor) ; Staněk, Jakub (referee)
This thesis concentrates on the Strong Law of Large Numbers. It features two proofs of this law. The first is less general, but simpler Borel's proof. The second one is more complex. It uses Kronecker's lemma and Kolmogorov-Khinchin's theorem, which is proven by Kolmogorov's inequality. The text includes all the necessary auxiliary theorems and lemmas along with their proofs. Since all the proofs are explored in a great detail this text is suitable for readers with only basic knowledge of probability theory and measure theory. Furthermore it contains numerous practical and mathematical examples thought out the whole text. Finally to demonstrate the importance of Strong Law of Large Numbers the text features four important applications of the law in mathematics.
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