Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Α. Γιαννόπουλος Καθηγητής ΕΚΠΑ (επιβλέπων), Α. Κατάβολος Καθηγητής ΕΚΠΑ, Δ. Χελιώτης Επίκουρος Καθηγητής ΕΚΠΑ
Original Title:
Κεντρικό οριακό πρόβλημα για λογαριθμικά κοίλα μέτρα πιθανότητας
Translated title:
Central limit problem for log-concave probability measures
Summary:
This thesis studies the geometry of log-concave probability measures and in
particular the central limit problem; this is the question to identify those
high-
dimensional distributions which have approximately Gaussian marginals. A
general fact states that if an isotropic probability measure is highly
concentrated
in a very thin shell then the answer to the question is a_rmative for almost all
one-dimensional marginals. Thus, the central limit problem is reduced to the
question of identifying those high-dimensional distributions that satisfy a thin
shell condition. It is important to mention that there exist isotropic
probability
measures for which this is not true. We show that the assumption of
log-concavity guarantees a thin shell bound, and hence an a_rmative answer to
the central limit problem. We present a omplete proof of the currently best
known estimate for the width of the thin shell, which is due to Gu_edon and E.
Milman.
Keywords:
Central limit problem, Log-concave measure, Isotropic measure
