Limit Distributions for Sums of Independent

Limit Distributions for Sums of Independent Random Vectors is a comprehensive reference that provides an up-to-date survey of the state of the art in this important research area.

MARK M. MEERSCHAERT, PhD, is Associate Professor of Mathematics at the University of NevadaùReno. He is also the author of Mathematical Modeling.

Preface

ix

Acknowledgments

xiii

Part I INTRODUCTION

Random Vectors

1

(18)

Probability Distributions

1

(7)

Convergence in Distribution

8

(6)

Characteristic Functions

14

(5)

Linear Operators

19

(18)

Operator Norms

19

(6)

Exponential Operators and Powers

25

(6)

Convergence of Types

31

(6)

Infinitely Divisible Distributions and Triangular Arrays

37

(58)

Infinitely Divisible Distributions

37

(15)

Convergence of Triangular Arrays

52

(20)

Domains of Attraction

72

(16)

Appendix: Continuous mappings into the circle

88

(3)

Notes and Comments

91

(4)

Part II MULTIVARIATE REGULAR VARIATION

Regular Variation for Linear Operators

95

(30)

Definitions and Basic Properties

95

(3)

Uniform Convergence and Path Behavior

98

(15)

The Spectral Decomposition

113

(10)

Notes and Comments

123

(2)

Regular Variation for Real-Valued Functions

125

(42)

Regularly Varying Functions

125

(6)

Exponents and Symmetries

131

(11)

Uniform Regular Variation

142

(23)

Notes and Comments

165

(2)

Regular Variation for Borel Measures

167

(82)

Regularly Varying Measures

167

(18)

R---O Varying Measures

185

(7)

Truncated Moments and Tail Moments

192

(35)

Sharp Spectral Bounds

227

(18)

Notes and Comments

245

(4)

Part III MULTIVARIATE LIMIT THEOREMS

The Limit Distributions

249

(40)

Operator Semistable Laws

249

(10)

Operator Stable Laws

259

(4)

Stable Laws

263

(13)

Semistable Laws

276

(3)

Structure Theorems

279

(6)

Notes and Comments

285

(4)

Central Limit Theorems

289

(56)

Normal Limits

289

(14)

Nonnormal Limits

303

(18)

General Limits

321

(21)

Stochastic Compactness

335

(3)

Operator Stable Limits

338

(4)

Notes and Comments

342

(3)

Related Limit Theorems

345

(32)

Large Deviations

345

(6)

Law of the Iterated Logarithm

351

(16)

Convergence of Semitypes

367

(7)

Notes and Comments

374

(3)

Part IV APPLICATIONS

Applications to Statistics

377

(76)

Sample Moments

377

(12)

Sample Covariance Matrix

389

(11)

Self-Normalized Sums

400

(5)

Tail and Moment Estimators

405

(11)

Symmetric k-Tensors

416

(8)

Time Series Analysis

424

(27)

Notes and Comments

451

(2)

Self-Similar Stochastic Processes

453

(14)

Operator Self-Similar Processes

453

(5)

Scaling Limits

458

(8)

Notes and Comments

466

(1)

References

467

(14)

Index

481

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Limit Distributions for Sums of Independent Random Vectors Heavy Tails in Theory and Practice

Summary

Author Biography

Table of Contents

Supplemental Materials