Information theory and the Central Limit Theorem:
Saved in:
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press
2004
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 209 S. graph. Darst. |
| ISBN: | 1860944736 9781860944734 |
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Record in the Search Index
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| adam_text | INFORMATION THEORY AND THE CENTRAL LIMIT THEOREM OLIVER JOHNSON
UNIVERSITY OF CAMBRIDGE, UK IMPERIAL COLLEGE PRESS CONTENTS PREFACE VII
1. INTRODUCTION TO INFORMATION THEORY 1 1.1 ENTROPY AND RELATIVE ENTROPY
1 1.1.1 DISCRETE ENTROPY 1 1.1.2 DIFFERENTIAL ENTROPY 5 1.1.3 RELATIVE
ENTROPY 8 1.1.4 OTHER ENTROPY-LIKE QUANTITIES 13 1.1.5 AXIOMATIC
DEFINITION OF ENTROPY 16 1.2 LINK TO THERMODYNAMIC ENTROPY 17 1.2.1
DEFINITION OF THERMODYNAMIC ENTROPY 17 1.2.2 MAXIMUM ENTROPY AND THE
SECOND LAW 19 1.3 FISHER INFORMATION 21 1.3.1 DEFINITION AND PROPERTIES
21 1.3.2 BEHAVIOUR ON CONVOLUTION 25 1.4 PREVIOUS INFORMATION-THEORETIC
PROOFS 27 1.4.1 RENYI S METHOD 27 1.4.2 CONVERGENCE OF FISHER
INFORMATION 30 2. CONVERGENCE IN RELATIVE ENTROPY 33 2.1 MOTIVATION 33
2.1.1 SANDWICH INEQUALITY 33 2.1.2 PROJECTIONS AND ADJOINTS 36 2.1.3
NORMAL CASE 38 2.1.4 RESULTS OF BROWN AND BARRON 41 2.2 GENERALISED
BOUNDS ON PROJECTION EIGENVALUES 43 XII INFORMATION THEORY AND THE
CENTRAL LIMIT THEOREM 2.2.1 PROJECTION OF FUNCTIONS IN L 2 43 2.2.2
RESTRICTED POINCARE CONSTANTS 44 2.2.3 CONVERGENCE OF RESTRICTED
POINCARE CONSTANTS 46 2.3 RATES OF CONVERGENCE 47 2.3.1 PROOF OF O(L/N)
RATE OF CONVERGENCE 47 2.3.2 COMPARISON WITH OTHER FORMS OF CONVERGENCE
50 2.3.3 EXTENDING THE CRAMER-RAO LOWER BOUND 51 3. NON-IDENTICAL
VARIABLES AND RANDOM VECTORS 55 3.1 NON-IDENTICAL RANDOM VARIABLES 55
3.1.1 PREVIOUS RESULTS 55 3.1.2 IMPROVED PROJECTION INEQUALITIES 57 3.2
RANDOM VECTORS 64 3.2.1 DEFINITIONS 64 3.2.2 BEHAVIOUR ON CONVOLUTION 65
3.2.3 PROJECTION INEQUALITIES 66 4. DEPENDENT RANDOM VARIABLES 69 4.1
INTRODUCTION AND NOTATION 69 4.1.1 MIXING COEFFICIENTS 69 4.1.2 MAIN
RESULTS 72 4.2 FISHER INFORMATION AND CONVOLUTION 74 4.3 PROOF OF
SUBADDITIVE RELATIONS 77 4.3.1 NOTATION AND DEFINITIONS 77 4.3.2 BOUNDS
ON DENSITIES 79 4.3.3 BOUNDS ON TAILS 82 4.3.4 CONTROL OF THE MIXING
COEFFICIENTS 83 5. CONVERGENCE TO STABLE LAWS 87 5.1 INTRODUCTION TO
STABLE LAWS 87 5.1.1 DEFINITIONS 87 5.1.2 DOMAINS OF ATTRACTION 89 5.1.3
ENTROPY OF STABLE LAWS 91 5.2 PARAMETER ESTIMATION FOR STABLE
DISTRIBUTIONS 92 5.2.1 MINIMISING RELATIVE ENTROPY 92 5.2.2 MINIMISING
FISHER INFORMATION DISTANCE 94 5.2.3 MATCHING LOGARITHM OF DENSITY 95
5.3 EXTENDING DE BRUIJN S IDENTITY 96 CONTENTS XIII 5.3.1 PARTIAL
DIFFERENTIAL EQUATIONS 96 5.3.2 DERIVATIVES OF RELATIVE ENTROPY 97 5.3.3
INTEGRAL FORM OF THE IDENTITIES 100 5.4 RELATIONSHIP BETWEEN FORMS OF
CONVERGENCE 102 5.5 STEPS TOWARDS A BROWN INEQUALITY 105 6. CONVERGENCE
ON COMPACT GROUPS 109 6.1 PROBABILITY ON COMPACT GROUPS 109 6.1.1
INTRODUCTION TO TOPOLOGICAL GROUPS 109 6.1.2 CONVERGENCE OF CONVOLUTIONS
ILL 6.1.3 CONDITIONS FOR UNIFORM CONVERGENCE 114 6.2 CONVERGENCE IN
RELATIVE ENTROPY 118 6.2.1 INTRODUCTION AND RESULTS 118 6.2.2 ENTROPY ON
COMPACT GROUPS 119 6.3 COMPARISON OF FORMS OF CONVERGENCE 121 6.4 PROOF
OF CONVERGENCE IN RELATIVE ENTROPY 125 6.4.1 EXPLICIT RATE OF
CONVERGENCE 125 6.4.2 NO EXPLICIT RATE OF CONVERGENCE 126 7. CONVERGENCE
TO THE POISSON DISTRIBUTION 129 7.1 ENTROPY AND THE POISSON DISTRIBUTION
129 7.1.1 THE LAW OF SMALL NUMBERS 129 7.1.2 SIMPLEST BOUNDS ON RELATIVE
ENTROPY 132 7.2 FISHER INFORMATION 136 7.2.1 STANDARD FISHER INFORMATION
136 7.2.2 SCALED FISHER INFORMATION 138 7.2.3 DEPENDENT VARIABLES 140
7.3 STRENGTH OF BOUNDS 142 7.4 DE BRUIJN IDENTITY 144 7.5 L BOUNDS ON
POISSON DISTANCE 146 7.5.1 L 2 DEFINITIONS 146 7.5.2 SUMS OF BERNOULLI
VARIABLES 147 7.5.3 NORMAL CONVERGENCE 150 8. FREE RANDOM VARIABLES 153
8.1 INTRODUCTION TO FREE VARIABLES 153 8.1.1 OPERATORS AND ALGEBRAS 153
8.1.2 EXPECTATIONS AND CAUCHY TRANSFORMS 154 XIV INFORMATION THEORY AND
THE CENTRAL LIMIT THEOREM 8.1.3 FREE INTERACTION 158 8.2 DERIVATIONS AND
CONJUGATE FUNCTIONS 161 8.2.1 DERIVATIONS 161 8.2.2 FISHER INFORMATION
AND ENTROPY 163 8.3 PROJECTION INEQUALITIES 166 APPENDIX A CALCULATING
ENTROPIES 171 A.I GAMMA DISTRIBUTION 171 A.2 STABLE DISTRIBUTIONS 173
APPENDIX B POINCARE INEQUALITIES 177 B.I STANDARD POINCARE INEQUALITIES
177 B.2 WEIGHTED POINCARE INEQUALITIES 179 APPENDIX C DE BRUIJN IDENTITY
183 APPENDIX D ENTROPY POWER INEQUALITY 187 APPENDIX E RELATIONSHIPS
BETWEEN DIFFERENT FORMS OF CONVERGENCE 191 E.I CONVERGENCE IN RELATIVE
ENTROPY TO THE GAUSSIAN 191 E.2 CONVERGENCE TO OTHER VARIABLES 194 E.3
CONVERGENCE IN FISHER INFORMATION 195 BIBLIOGRAPHY 199 INDEX 207
|
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| spelling | Johnson, Oliver Verfasser (DE-588)142485616 aut Information theory and the Central Limit Theorem Oliver Johnson London Imperial College Press 2004 XIV, 209 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Informatietheorie gtt Information, Théorie de l' - Méthodes statistiques Limiettheorema's gtt Probabilités Théorème central limite Central limit theorem Information theory Statistical methods Probabilities Zufallsvariable (DE-588)4129514-6 gnd rswk-swf Informationstheorie (DE-588)4026927-9 gnd rswk-swf Zufallsvektor (DE-588)4191098-9 gnd rswk-swf Informationstheorie (DE-588)4026927-9 s DE-604 Zufallsvektor (DE-588)4191098-9 s Zufallsvariable (DE-588)4129514-6 s HEBIS Datenaustausch Mainz application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013071688&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Johnson, Oliver Information theory and the Central Limit Theorem Informatietheorie gtt Information, Théorie de l' - Méthodes statistiques Limiettheorema's gtt Probabilités Théorème central limite Central limit theorem Information theory Statistical methods Probabilities Zufallsvariable (DE-588)4129514-6 gnd Informationstheorie (DE-588)4026927-9 gnd Zufallsvektor (DE-588)4191098-9 gnd |
| subject_GND | (DE-588)4129514-6 (DE-588)4026927-9 (DE-588)4191098-9 |
| title | Information theory and the Central Limit Theorem |
| title_auth | Information theory and the Central Limit Theorem |
| title_exact_search | Information theory and the Central Limit Theorem |
| title_full | Information theory and the Central Limit Theorem Oliver Johnson |
| title_fullStr | Information theory and the Central Limit Theorem Oliver Johnson |
| title_full_unstemmed | Information theory and the Central Limit Theorem Oliver Johnson |
| title_short | Information theory and the Central Limit Theorem |
| title_sort | information theory and the central limit theorem |
| topic | Informatietheorie gtt Information, Théorie de l' - Méthodes statistiques Limiettheorema's gtt Probabilités Théorème central limite Central limit theorem Information theory Statistical methods Probabilities Zufallsvariable (DE-588)4129514-6 gnd Informationstheorie (DE-588)4026927-9 gnd Zufallsvektor (DE-588)4191098-9 gnd |
| topic_facet | Informatietheorie Information, Théorie de l' - Méthodes statistiques Limiettheorema's Probabilités Théorème central limite Central limit theorem Information theory Statistical methods Probabilities Zufallsvariable Informationstheorie Zufallsvektor |
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