Geometry and martingales in Banach spaces PDF
Preview Geometry and martingales in Banach spaces
(cid:105) (cid:105) “K389703˙FM” — 2018/8/16 — 16:04 — page 2 — #2 (cid:105) (cid:105) Geometry and Martingales in Banach Spaces (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) “K389703˙FM” — 2018/8/16 — 16:04 — page 3 — #3 (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) “K389703˙FM” — 2018/8/16 — 16:04 — page 4 — #4 (cid:105) (cid:105) Geometry and Martingales in Banach Spaces Wojbor A. Woyczyński Case Western Reserve University (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) (cid:105) “K389703˙FM” — 2018/8/16 — 16:04 — page 6 — #6 (cid:105) (cid:105) CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20180816 International Standard Book Number-13: 978-1-138-61637-0 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Woyczyânski, W. A. (Wojbor Andrzej), 1943- author. Title: Geometry and martingales in Banach spaces / Wojbor A. Woyczynski (Case Western Reserve University). Description: Boca Raton, Florida : CRC Press, [2018] | Includes bibliographical references and index. Identifiers: LCCN 2018026561| ISBN 9781138616370 (hardback : alk. paper) | ISBN 9780429462153 (ebook) Subjects: LCSH: Martingales (Mathematics) | Geometric analysis. | Banach spaces. Classification: LCC QA274.5 .W69 2018 | DDC 519.2/36--dc23 LC record available at https://lccn.loc.gov/2018026561 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com (cid:105) (cid:105) (cid:105) (cid:105) ✐ ✐ “A˙Book” — 2018/8/22 — 11:55 — page v — #2 ✐ ✐ Contents Introduction ix 1 Preliminaries: Probability and geometry in Banach spaces 1 1.1 Random vectors in Banach spaces . . . . . . . . . . 1 1.2 Random series in Banach spaces . . . . . . . . . . . 3 1.3 Basic geometry of Banach spaces . . . . . . . . . . 8 1.4 Spaces with invariant under spreading norms which are finitely representable in a given space . . . . . . 13 1.5 Absolutely summing operators and factorization results . . . . . . . . . . . . . . . 16 2 Dentability, Radon-Nikodym Theorem, and Mar- tingale Convergence Theorem 23 2.1 Dentability . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Dentability versus Radon-Nikodym property, and martingale convergence . . . . . . . . . . . . . . . . 29 2.3 Dentability and submartingales in Banach lattices and lattice bounded operators . . . . . . . . . . . . 38 3 Uniform Convexity and Uniform Smoothness 47 3.1 Basic concepts . . . . . . . . . . . . . . . . . . . . 47 3.2 Martingales in uniformly smooth and uniformly convex spaces . . . . . . . . . . . . . 51 3.3 General concept of super-property . . . . . . . . . . 61 3.4 Martingales in super-reflexive Banach spaces . . . . 63 v ✐ ✐ ✐ ✐ ✐ ✐ “A˙Book” — 2018/8/22 — 11:55 — page vi — #3 ✐ ✐ vi CONTENTS 4 Spaces that do not contain c 67 0 4.1 Boundedness and convergence of random series . . . 67 4.2 Pre-Gaussian random vectors . . . . . . . . . . . . 72 5 Cotypes of Banach spaces 75 5.1 Infracotypes of Banach spaces . . . . . . . . . . . . 75 5.2 Spaces of Rademacher cotype . . . . . . . . . . . . 79 5.3 Local structure of spaces of cotype q . . . . . . . . 85 5.4 Operators in spaces of cotype q . . . . . . . . . . . 92 5.5 Random series and law of large numbers . . . . . . 99 5.6 Central limit theorem, law of the iterated loga- rithm, and infinitely divisible distributions . . . . . 110 6 Spaces of Rademacher and stable types 115 6.1 Infratypes of Banach spaces . . . . . . . . . . . . . 115 6.2 Banach spaces of Rademacher-type p . . . . . . . . 119 6.3 Local structures of spaces of Rademacher-type p . . 131 6.4 Operators on Banach spaces of Rademacher-type p 140 6.5 Banach spaces of stable-type p and their local structures . . . . . . . . . . . . . . . 144 6.6 Operators on spaces of stable-type p . . . . . . . . 153 6.7 Extented basic inequalities and series of random vectors in spaces of type p . . . . . 159 6.8 Strong laws of large numbers and asymptotic be- havior of random sums in spaces of Rademacher- type p . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.9 Weak and strong laws of large numbers in spaces of stable-type p . . . . . . . . . . . . . . . . . . . . . . 178 6.10 Random integrals, convergence of infinitely divisi- ble measures and the central limit theorem . . . . . 182 7 Spaces of type 2 197 7.1 Additional properties of spaces of type 2 . . . . . . 197 7.2 Gaussian random vectors . . . . . . . . . . . . . . . 202 7.3 Kolmogorov’s inequality and three-series theorem . 206 7.4 Central limit theorem . . . . . . . . . . . . . . . . . 208 7.5 Law of iterated logarithm . . . . . . . . . . . . . . 218 ✐ ✐ ✐ ✐ ✐ ✐ “A˙Book” — 2018/8/22 — 11:55 — page vii — #4 ✐ ✐ CONTENTS vii 7.6 Spaces of type 2 and cotype 2 . . . . . . . . . . . . 223 8 Beck convexity 227 8.1 General definitions and properties and their rela- tionship to types of Banach spaces . . . . . . . . . 227 8.2 LocalstructureofB-convex spaces andpreservation of B-convexity under standard operations . . . . . . 236 8.3 Banach lattices and reflexivity of B-convex spaces . . . . . . . . . . . . . . . . . . 242 8.4 Classical weak and strong laws of large numbers in B-convex spaces . . . . . . . . . . . . . . . . . . . . 249 8.5 Laws of large numbers for weighted sums and not necessarily independent summands . . . . . . . . . 258 8.6 Ergodic properties of B-convex spaces . . . . . . . . 263 8.7 Trees in B-convex spaces . . . . . . . . . . . . . . . 271 9 Marcinkiewicz-Zygmund Theorem in Banach spaces 273 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . 273 9.2 Brunk-Prokhorov’s type strong law and related rates of convergence . . . . . . . . . . . . . . . . . . 276 9.3 Marcinkiewicz-Zygmund type strong law and re- lated rates of convergence . . . . . . . . . . . . . . 279 9.4 Brunk and Marcinkiewicz-Zygmund type strong laws for martingales . . . . . . . . . . . . . . . . . . 288 Bibliography 297 Index 313 ✐ ✐ ✐ ✐ ✐ ✐ “A˙Book” — 2018/8/22 — 11:55 — page viii — #5 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ “A˙Book” — 2018/8/22 — 11:55 — page ix — #6 ✐ ✐ Introduction Inthisvolume weareproviding acompact expositionoftheresults explaining the interrelations existing between the metric geome- try of Banach spaces and probability theory of random vectors with values in those Banach spaces. In particular martingales and random series of independent random vectors are studied. Chapter1introducesthebasicgeometricandprobabilisticcon- cepts in Banach spaces. Chapter 2 concentrates on the geometric concept of dentability and provides an exposition of the results originally due to M.A. Rieffel, H.B. Maynard, S.D. Chatterjii, and others. The concept of dentability turns out to be very natural in thecontextofmartingaleseventhoughitwasoriginallyintroduced in a study of the Radon-Nikodym theorem in Banach spaces. The chapter ends with an exposition of the theory of sub-martingales with values in Banach lattices and the related issues of the lattice bounded operators. Chapter 3 deals with the two classical concepts of metric ge- ometry in Banach spaces, namely, the uniform smoothness and the uniform convexity. Here, the works of G. Pisier and P. Assuad (also, see the important 560-page long recent monograph, Mar- tingales in Banach Spaces, by G. Pisier) showed that some of the results obtained earlier by the author for sums of independent ran- dom vectors in such Banach spaces carry over to the more general situationofmartingales, andevenprovideacompletecharacteriza- tion of those geometric properties in the language of martingales. Finite tree property and super-reflexivity, the notions introduced by R.C. James, turn out to be the properties that are most inti- mately related to the martingale theory as shown by results of S. Kwapien´, and G. Pisier, which are discussed in this chapter. ix ✐ ✐ ✐ ✐
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