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Hájek-Renyi inequality Bělohlávek, Ivan ; Prášková, Zuzana (advisor) ; Čoupek, Petr (referee) Title: A Hájek-Renyi inequality Author: Ivan Bělohlávek Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Zuzana Prášková, CSc., Department of Probability and Mathematical Statistics Abstract: In this thesis we study the Hájek-Rényi inequality for mixingales and their special cases. First, we prove the Hájek-Rényi inequality for martingales. Then, we investigate the relationship between the Kolomogorov and Hájek-Rényi inequalities. After that, we prove the law of large numbers using the Hájek-Rényi inequality. We then provide a detailed proof of maximal inequality for mixingales, which we then use to derive the Hájek-Rényi inequality for mixingales. We then apply the inequality to a multitude of special cases of mixingales. Keywords: Hájek-Rényi inequality, martingales, mixingales, linear process 1 Detailed record

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