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1	The law of the iterated logarithm for tail sums Ghimire, Santosh January 1900 (has links) Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis. The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series. We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm. We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it. Sum of Rademacher functions is a nicely behaved dyadic martingale. With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales. We obtain both upper and lower bound in the case of martingales. A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables. Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables. So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund. Law of the iterated logarithm dyadic martingales Lacunary series Rademacher functions tail law of the iterated logarithm Mathematics (0405)
2	Arithmetic Properties of Values of Lacunary Series Bradshaw, Ryan 12 September 2013 (has links) A lacunary series is a Taylor series with large gaps between its non-zero coefficients. In this thesis we exploit these gaps to obtain results of linear independence of values of lacunary series at integer points. As well, we will study different methods found in Diophantine approximation which we use to study arithmetic properties of values of lacunary series at algebraic points. Among these methods will be Mahler's method and a new approach due to Jean-Paul Bézivin. Lacunary Series Linear independence Fredholm Series Mahler's Method Method of Bezivin The Gap Method
3	Arithmetic Properties of Values of Lacunary Series Bradshaw, Ryan January 2013 (has links) A lacunary series is a Taylor series with large gaps between its non-zero coefficients. In this thesis we exploit these gaps to obtain results of linear independence of values of lacunary series at integer points. As well, we will study different methods found in Diophantine approximation which we use to study arithmetic properties of values of lacunary series at algebraic points. Among these methods will be Mahler's method and a new approach due to Jean-Paul Bézivin. Lacunary Series Linear independence Fredholm Series Mahler's Method Method of Bezivin The Gap Method
4	Sur la dimension de Minkowski des quasicercles / On Minkowski dimension of quasicircles Le, Thanh Hoang Nhat 05 October 2012 (has links) Pour accéder au résumé en français à la fin de la thèse, ouvrir le fichier du texte intégral / Pour accéder au résumé en anglais à la fin de la thèse, ouvrir le fichier du texte intégral Dimension de Minkowski Dimension de Hausdorff Fonction de Bloch Question ouverte de Mc Mullen Martingale dyadique Mesure de Kahane Série lacunaire Quasicercles Minkowski dimension Hausdorff dimension Bloch function Mc Mullen’s open question Dyadic martingale Kahane measure Lacunary series Quasicircles

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