an:05295871
Zbl 1152.60025
Collet, Pierre
Some aspects of the central limit theorem and related topics
EN
Fournier, Jean-Daniel (ed.) et al., Harmonic analysis and rational approximation. Their r??les in signals, control and dynamical systems. Berlin: Springer (ISBN 3-540-30922-5/pbk). Lecture Notes in Control and Information Sciences 327, 105-127 (2006).
2006
a
60F05 60F15 28A78 37C45 60F10
central limit theorem; independent random variables; Brownian motion; strong invariance principle; large deviations; multifractal measures; Hausdorff dimension
The paper provides a concise overview of probabilistic phenomena around the central limit theorem. Begining from the game of head and tail and De Moivre-Laplace theorem through classical L??vy, Lindeberg, Feller and Berry-Ess??en theorems up to the law of the iterated logarithm and the strong invariance principle the author gives an elementary and clear introduction to the subject. Two last sections of the paper are devoted to more advanced questions. In the first of them basic large deviations theorem is presented and an application to the game of head and tail is shown in detail. In the last section, elaborated most extensively, the multifractal analysis of a measure is exposed including notions of monofractal measures and the Hausdorff dimension. Particular attention is payed to an interesting and relatively simple case of measures defined on the Cantor set. The paper is addressed to nonspecialists and is written in a very accessible way. Extensive and carefully selected bibliography can help the reader along a deeper study.
For the entire collection see [Zbl 1083.42001].
Tadeusz Inglot (Wroc??aw)