Sums and Gaussian vectors:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1995
|
| Series: | Lecture notes in mathematics
1617 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 305 S. |
| ISBN: | 3540603115 9783540603115 |
Staff View
MARC
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| 245 | 1 | 0 | |a Sums and Gaussian vectors |c Vadim Yurinsky |
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| 490 | 1 | |a Lecture notes in mathematics |v 1617 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Gaussian Measures in Euclidean Space 1
1.1 Preliminaries 1
1.1.1 Standard Normal Distribution 1
1.1.2 Gaussian Distributions: Basic Definitions 2
1.1.3 Characterization Theorems 4
1.1.4 Monotonicity in Covariances 4
1.1.5 Conditional Distributions and Projections 5
1.1.6 Laplace Transform for DF of Squared Norm 6
1.2 Extremal Properties of Half Spaces 7
1.2.1 Isoperimetric Property of Sphere 7
1.2.2 Constructing Gaussian Law from Uniform Ones 8
1.2.3 An Isoperimetric Inequality for Gaussian Laws 9
1.2.4 Application to Large Deviations 11
1.3 Dilations of Convex Sets 11
1.3.1 Some Special Symmetric Densities 11
1.3.2 Majoration by Measures of Half Spaces 14
1.3.3 Isoperimetric Inequalities for Convex Sets 18
1.4 Some Formulae for Convex Functions 20
1.4.1 Special Coordinates 20
1.4.2 Volume Integrals over Complements to Level Sets 21
1.4.3 Derivatives of Integrals over Level Sets 22
1.4.4 Piecewise Linear Functions and Poly topes 24
1.4.5 Integral Representation for Distribution of CPLF 27
1.4.6 Approximation of Piecewise Linear Functions 29
1.5 Smoothness of Gaussian CPLF Tails 30
1.5.1 Tail Density of CPLF: Existence and Lower Bound .. 30
1.5.2 Bounds for Inverses of DF s 32
1.5.3 Upper Bounds for Density 33
1.5.4 Derivation of Polynomial Bound for Density 34
1.5.5 Derivation of Exponential Bound for Density 35
1.5.6 Bounds on Upper Derivatives of CPLF Density 36
1.5.7 Estimates for Derivative of Density: Smooth Case .... 37
1.5.8 Evaluation of Upper Derivative: CPLF Density 39
1.6 Some Related Results 40
Vlll CONTENTS
2 Seminorms of Gaussian Vectors in Infinite Dimensions 43
2.1 Exponential Summability of Seminorms 43
2.1.1 Summability of Quadratic Exponential 43
2.1.2 Proof of Summability of Quadratic Exponential 45
2.1.3 Highest Order of Finite Exponential Moment 46
2.2 Suprema of Gaussian Sequences 49
2.2.1 Absolute Continuity of Tail 49
2.2.2 Properties of Density: Upper Bounds 50
2.2.3 Examples of Discontinuities 51
2.2.4 Absolute Continuity and Upper Bounds: Proofs 52
2.2.5 Monotonicity of Inverse DF Ratio 54
2.2.6 Sequences with Constant Variance 54
2.2.7 Large Deviation Asymptotics for Density 56
2.3 Norm of Gaussian RV in Hilbert Space 58
2.3.1 Asymptotic Expressions for Tails of Norm 58
2.3.2 An Auxiliary Measure 60
2.3.3 Derivation of Asymptotic Formulae 62
2.3.4 Direct Derivation of Large Deviation Asymptotics .... 64
2.4 Probabilities of Landing in Spheres 65
2.4.1 Asymptotics for Small and Displaced Spheres .... 65
2.4.2 Laplace Transform of Squared Norm Density 67
2.4.3 Derivation of Density Asymptotics 68
2.4.4 Distant Ball Asymptotics 71
2.5 Related Results 73
2.5.1 Geometric Theorems 73
2.5.2 Theorems on Gaussian Processes 76
3 Inequalities for Seminorms:
Sums of Independent Random Vectors 79
3.1 Bounds on Tails of Maximal Sums 80
3.1.1 Relations Between Maximal Sums 80
3.1.2 Maximal Sums and Maximal Summands 81
3.1.3 Case of Polynomial Tail of Maximal Summand 82
3.1.4 Bounds on Moments of Maximal Sum 84
3.1.5 Exponential Moments of Maximal Sums 85
3.2 Bounds Based on Symmetry 86
3.2.1 The Levy Inequality 86
3.2.2 Symmetrization 87
3.2.3 A Chebyshev Type Bound 88
3.3 Exponential Bounds for Sums 90
3.3.1 S. N. Bernstein s Bound 90
3.3.2 Exponential Bounds for Arbitrary Seminorms 91
3.3.3 Exponential Inequalities for Banach Space 93
3.3.4 Exponential Inequalities for Hilbert Space 94
3.4 Applications of Exponential Bounds 97
3.4.1 An Asymptotic Formula for Large Deviations 97
CONTENTS IX
3.4.2 Bounds on Exponential Moments 100
3.5 Bounds Using Finite Order Moments 102
3.5.1 Nagaev Fuk Inequality 102
3.5.2 Demonstration of Nagaev Fuk Inequality 103
3.5.3 Inequalities for Moments 106
3.5.4 Estimates in Terms of Sums Without Repetitions 107
3.6 The Cramer Transform Ill
3.6.1 Definition and Inversion Formula Ill
3.6.2 Moments of the Cramer Conjugate 113
3.7 Truncations 117
3.7.1 Inequalities for Probabilities 117
3.7.2 Auxiliary Inequalities for Moments 119
3.7.3 Derivation of Truncation Inequalities 120
4 Rough Asymptotics of Large Deviations 123
4.1 Deviation Functional and Deviation Rates 123
4.1.1 Definitions and Main Results 123
4.1.2 The Neighborhood Deviation Rate 125
4.1.3 DF and Half Space Deviation Rate 128
4.1.4 Identity of DF and Local Rates 130
4.2 Information Distance 132
4.2.1 Definition of the Information Distance 132
4.2.2 An Extremal Property of Information Distance 134
4.3 Information Distance and DF 139
4.3.1 Identity of DF and Minimal Information Distance .... 139
4.3.2 Identification of DF and Information Distance 140
4.4 LD Asymptotics for Sums in Banach Space 143
4.4.1 LD Asymptotics and Deviation Functional 143
4.4.2 Calculation of Probability Asymptotics 143
4.4.3 Asymptotics of Exponential Moments 146
4.5 Rough Gaussian LD Asymptotics 148
4.5.1 Asymptotic Expressions for LD Probabilities 148
4.5.2 Reduction to LD Asymptotics for Sums 150
4.6 Moderately Large Deviations for Sums 153
4.6.1 Asymptotic Formulae 153
4.6.2 An Auxiliary Result 154
4.6.3 Calculation of Asymptotics 155
4.6.4 Moderately Large Deviations in Banach Space 158
5 Gaussian and Related Approximations
for Distributions of Sums 163
5.1 Operator Approach to Limit Theorems 164
5.1.1 Translations and Derivatives 164
5.1.2 Average Translations and Their Expansions 166
5.1.3 Operator Interpretation of Classical Limit Theorems . . . 168
5.2 Asymptotic Expansions for Expectations of Smooth Functions . . 169
X CONTENTS
5.2.1 Heuristic Derivation of Asymptotic Expansions 169
5.2.2 Asymptotic Expansion for Individual Summand 171
5.2.3 Bergstrom and Edgeworth Type Expansions 173
5.3 Smooth Mappings to Finite Dimensions 177
5.3.1 Approximation via Asymptotic Expansion for CF 177
5.3.2 Smoothing Inequality and Fourier Transforms 179
5.4 Fourier Transforms Near the Origin 181
5.4.1 Regularity Assumptions 182
5.4.2 Bounds on Remainder Terms 185
5.5 Reduction to Pure CF 188
5.5.1 Notation and Restrictions 188
5.5.2 Reduction to Polynomial Imaginary Exponent 189
5.5.3 Products of Polynomials with Imaginary Exponentials . . 191
5.6 Almost Linear Mappings 193
5.6.1 Gaussian Case 193
5.6.2 Sums in the Quasi Gaussian Range 199
5.6.3 Central Limit Theorem with Error Estimate 205
5.7 Bounds on CF of Squared Norm 209
5.7.1 A Symmetrization Inequality for CF s 209
5.7.2 Bounds on CF of Squared Norm for Sums 210
5.7.3 Error Estimates for Higher Order Expansions 214
6 Fine Asymptotics of Moderate Deviations 217
6.1 Fine LD Asymptotics in the Gaussian Case 218
6.1.1 Asymptotic Expression for Probability 218
6.1.2 The Laplace Method: a Reminder 223
6.1.3 Derivation of LD Asymptotics 226
6.2 Landing Outside a Sphere 229
6.2.1 Constructing the Focusing Device 230
6.2.2 A Correct Bernstein Type Bound 233
6.2.3 Focusing and Cramer s Transform 235
6.2.4 Rough Exponential Bounds for Cramer Transforms .... 237
6.2.5 Derivation of Correct Bernstein Type Bound 240
6.3 Moderately Large Deviations for Sums 244
6.3.1 Focusing and Choosing Cramer Transform 244
6.3.2 The Cramer Series 247
6.3.3 The Normal Approximation 250
6.3.4 Solvability of Equations for Cramer Parameter 252
A Auxiliary Material 255
A.I Euclidean Space 255
A.I.I Vectors and Sets 255
A.1.2 Volume Integrals 256
A.1.3 Surface Integrals 256
A.2 Infinite Dimensional Random Elements 257
A.2.1 Random Elements in a Measurable Space 258
CONTENTS XI
A.2.2 Distributions in Polish Spaces 259
A.2.3 Measurable Linear Spaces 259
A.2.4 Cylinder Sets 260
A.2.5 Distributions in Banach Spaces 263
A.2.6 Distributions in Hilbert Space 266
A.3 Deviation Function of Real Random Variable 267
A.3.1 Large Deviations and Deviation Function 267
A.3.2 Deviation Function and Information 272
B Bibliographic Commentary 275
Basic Abbreviations and Notation 284
Bibliography 285
Index 304
|
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| dewey-ones | 510 - Mathematics |
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| id | DE-604.BV010359456 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:51:10Z |
| institution | BVB |
| isbn | 3540603115 9783540603115 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006897300 |
| oclc_num | 246274408 |
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| physical | XI, 305 S. |
| publishDate | 1995 |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Jurinskij, Vadim V. 1945- Verfasser (DE-588)104163054 aut Sums and Gaussian vectors Vadim Yurinsky Berlin [u.a.] Springer 1995 XI, 305 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1617 Gaussian sums Limit theorems (Probability theory) Grenzwertsatz (DE-588)4158163-5 gnd rswk-swf Zufallsvektor (DE-588)4191098-9 gnd rswk-swf Gauß-Summe (DE-588)4156109-0 gnd rswk-swf Große Abweichung (DE-588)4330658-5 gnd rswk-swf Gauß-Summe (DE-588)4156109-0 s DE-604 Zufallsvektor (DE-588)4191098-9 s Grenzwertsatz (DE-588)4158163-5 s Große Abweichung (DE-588)4330658-5 s Lecture notes in mathematics 1617 (DE-604)BV000676446 1617 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006897300&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jurinskij, Vadim V. 1945- Sums and Gaussian vectors Lecture notes in mathematics Gaussian sums Limit theorems (Probability theory) Grenzwertsatz (DE-588)4158163-5 gnd Zufallsvektor (DE-588)4191098-9 gnd Gauß-Summe (DE-588)4156109-0 gnd Große Abweichung (DE-588)4330658-5 gnd |
| subject_GND | (DE-588)4158163-5 (DE-588)4191098-9 (DE-588)4156109-0 (DE-588)4330658-5 |
| title | Sums and Gaussian vectors |
| title_auth | Sums and Gaussian vectors |
| title_exact_search | Sums and Gaussian vectors |
| title_full | Sums and Gaussian vectors Vadim Yurinsky |
| title_fullStr | Sums and Gaussian vectors Vadim Yurinsky |
| title_full_unstemmed | Sums and Gaussian vectors Vadim Yurinsky |
| title_short | Sums and Gaussian vectors |
| title_sort | sums and gaussian vectors |
| topic | Gaussian sums Limit theorems (Probability theory) Grenzwertsatz (DE-588)4158163-5 gnd Zufallsvektor (DE-588)4191098-9 gnd Gauß-Summe (DE-588)4156109-0 gnd Große Abweichung (DE-588)4330658-5 gnd |
| topic_facet | Gaussian sums Limit theorems (Probability theory) Grenzwertsatz Zufallsvektor Gauß-Summe Große Abweichung |
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