Associated Sequences, Demimartingales and Nonparametric Inference
This book gives a comprehensive review of results for associated sequences and demimartingales developed so far, with special emphasis on demimartingales and related processes. One of the basic aims of theory of probability and statistics is to buil
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B.L.S. Prakasa Rao
Associated Sequences, Demimartingales and Nonparametric Inference
B.L.S. Prakasa Rao Department of Mathematics and Statistics University of Hyderabad Hyderabad, India
ISBN 978-3-0348-0239-0 e-ISBN 978-3-0348-0240-6 DOI 10.1007/978-3-0348-0240-6 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2011941605 Mathematics Subject Classification (2010): 60G48 © Springer Basel AG 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)
In memory of my mother Bhagavatula Saradamba (1926–2002) for her abiding trust in me and for her love and affection
Contents Preface
xi
1 Associated Random Variables and Related Concepts 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some Probabilistic Properties of Associated Sequences . . . . . . . 1.3 Related Concepts of Association . . . . . . . . . . . . . . . . . . .
1 1 5 23
2 Demimartingales 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Characteristic Function Inequalities . . . . . . . . . . . . . . . 2.3 Doob Type Maximal Inequality . . . . . . . . . . . . . . . . . 2.4 An Upcrossing Inequality . . . . . . . . . . . . . . . . . . . . 2.5 Chow Type Maximal Inequality . . . . . . . . . . . . . . . . . 2.6 Whittle Type Maximal Inequality . . . . . . . . . . . . . . . . 2.7 More on Maximal Inequalities . . . . . . . . . . . . . . . . . . 2.8 Maximal φ-Inequalities for Nonnegative Demisubmartingales 2.9 Maximal Inequalities for Functions of Demisubmartingales . . 2.10 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . 2.11 Dominated Demisubmartingales . . . . . . . . . . . . . . . . .
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31 31 37 39 42 44 50 54 58 65 69 70
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73 73 79 81 86 89 92 94 95 98 102
3 N -Demimartingales 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.2 Maximal Inequalities . . . . . . . . . . . . . . 3.3 More on Maximal Inequalities . . . . . . . . . 3.4 Downcrossing Inequality . . . . . . . . . . . . 3.5 Chow Type Maximal Inequality . . . . . . . . 3.6 Functions of N -Demimartingales . . . . . . . 3.7 Strong Law of Large Numbers . . . . . . . . . 3.8 Azuma Type Inequality . . . . . . . . . . . . 3.9 Marcinkiewicz-Zygmund Type inequality . . . 3.10 Comparison Theorem on Moment Inequalities
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