Non-classical error bounds in the central limit theorem [Elektronische Ressource] / von Larisa Yaroslavtseva

Non-classical Error Bounds in the Central Limit TheoremDissertationzur Erlangung des akademischen Gradesdoctor rerum naturalium(Dr. rer. nat.)von Dipl. Math. Larisa Yaroslavtseva(akademischer Grad, Vorname, Name)geb. am 25.11.1983 in Miass/Russische F orderationgenehmigt durch die Facult at fur Mathematikder Otto-von-Guericke-Universit at MagdeburgGutachter: Prof. Dr. Ulyanov, Vladimir V.(akademischer Grad, Vorname, Name)Prof. Dr. Christoph, Gerd(akademischer Grad, Vorname, Name)eingereicht am: 25.05.2008Verteidigung am: 17.07.2008PrefaceThe classical central limit theorem states the uniform convergence of the dis-tribution functions of the standardized sums of independent and identicallydistributed square integrable real-valued random variables to the standardnormal distribution function. While rst versions of the central limit theoremare already due to Moivre (1730) and Laplace (1812), a systematic study ofthis topic started at the beginning of the last century with the fundamentalwork of Lyapunov (1900, 1901). Meanwhile, extensions of the central limittheorem are available for a multitude of settings. This includes, e.g., Ba-nach space valued random variables as well as substantial relaxations of theassumptions of independence and identical distributions. Furthermore, ex-plicit error bounds are established and asymptotic expansions are employedto obtain better approximations.