Postgraduate Thesis uoadl:1318129 508 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2013-02-22

2013

Χιώνη Λαμπρινή

Α. Γιαννόπουλος Καθηγητής ΕΚΠΑ (επιβλέπων), Α. Κατάβολος Καθηγητής ΕΚΠΑ, Δ. Χελιώτης Επίκουρος Καθηγητής ΕΚΠΑ

Κεντρικό οριακό πρόβλημα για λογαριθμικά κοίλα μέτρα πιθανότητας

Greek

Central limit problem for log-concave probability measures

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.

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.

Central limit problem, Log-concave measure, Isotropic measure

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