History of the Central Limit Theorem: From Classical to Modern Probability Theory

Cover......Page 1
Sources and Studies in the History of Mathematics and Physical Sciences......Page 2
A History of the Central Limit Theorem: From Classical to Modern Probability Theory......Page 4
9780387878560......Page 5
Preface......Page 6
Contents......Page 8
List of Figures......Page 14
Abbreviations and Denotations......Page 16
1 Introduction......Page 18
1.1 Different Versions of Central Limit Theorems......Page 20
1.2 Objectives and Focus of the Present Examination......Page 22
1.3 The Development of Analysis in the 19th Century......Page 26
1.4 Literature on the History of the Central Limit Theorem......Page 28
1.5 Terminology and Notation......Page 29
1.6 The Prehistory: De Moivre's Theorem......Page 31
2 The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods......Page 34
2.1 Laplace's Central "Limit'' Theorem......Page 35
2.1.1 Sums of Independent Random Variables......Page 36
2.1.2 Laplace's Method of Approximating Integrals, and "Algebraic Analysis''......Page 37
2.1.3 The Emergence of Characteristic Functions and the Deduction of Approximating Normal Distributions......Page 38
2.1.4 The "Rigor'' of Laplace's Analysis......Page 40
2.1.5.1 The Comet Problem......Page 42
2.1.5.2 The Foundation of the Method of Least Squares......Page 43
2.1.5.3 Benefits from Games of Chance......Page 47
2.2.1 Poisson's Concept of Random Variable......Page 48
2.2.2 Poisson's Representation of the Probabilities of Sums......Page 49
2.2.3.1 Poisson's Version of the Central Limit Theorem......Page 50
2.2.3.2 Poisson's Law of Large Numbers......Page 52
2.2.4 Poisson's Infinitistic Approach......Page 53
2.2.5 Approximation by Series Expansions......Page 56
2.3.1 Toward a New Conception of Mathematics......Page 57
2.3.3 The Rigorization of Laplace's Idea of Approximation......Page 59
2.4.1 Dirichlet's Modification of the Laplacian Method of Approximation......Page 61
2.4.2 The Application of the Discontinuity Factor......Page 63
2.4.3 Dirichlet's Proof......Page 64
2.4.3.2 Dirichlet's Discussion of the Limit......Page 65
2.5.1 The Cauchy--Bienaymé Dispute......Page 69
2.5.2 Cauchy's Exceptional Laws of Error......Page 72
2.5.3 Bienaymé's Arguments......Page 76
2.5.4 Cauchy's Version of the Central Limit Theorem......Page 78
2.5.5 Cauchy's Idea of Proof......Page 80
2.5.6 The End of the Controversy......Page 82
2.5.7 Conclusion: Steps Toward Modern Probability......Page 84
Appendix: Original Text of Dirichlet's Proof of the Central Limit Theorem According to Lecture Notes from 1846......Page 86
3 The Hypothesis of Elementary Errors......Page 92
3.1 Gauss and "His'' Error Law......Page 93
3.2 Hagen, Bessel, and "elementäre Fehler''......Page 96
3.2.1 The Rediscovery of the Hypothesis of Elementary Errors by Gotthilf Hagen......Page 97
3.2.2 Bessel's Generalization of the Hypothesis of Elementary Errors......Page 104
3.3.1 Normal Distributions in Statistics of Biological and Social Phenomena......Page 110
3.3.2 Advancement Within Error Theory......Page 112
3.3.2.1 Rectangularly Distributed Elementary Errors......Page 113
3.3.2.2 Crofton's Hypothesis......Page 115
3.3.2.3 Pizzetti's Account on the Hypothesis of Elementary Errors......Page 119
3.3.2.4 Schols, and Elementary Errors in Plane and Space......Page 121
3.4 Nonnormal Distributions, Series Expansions, and Modifications of the Hypothesis of Elementary Errors......Page 124
3.4.1 Approximations of "Arbitrary'' Probability Functions by Series in Hermite Polynomials......Page 126
3.4.2 The "Natural'' Role of the Normal Distribution and Its Derivatives......Page 132
3.4.2.1 Hausdorff's "Kanonische Parameter''......Page 133
3.4.2.2 Charlier's A Series......Page 136
3.4.2.3 Edgeworth and "The'' Law of Error......Page 139
3.4.3 The Method of Translation......Page 149
3.4.3.1 The Log-Normal Distribution......Page 150
3.4.3.2 Wicksell's General Model of Elementary Errors......Page 152
3.4.3.3 The Further Fate of the Hypothesis of Elementary Errors......Page 153
Appendix: Letter from Bessel to Jacobi, 14 August 1834......Page 155
4 Chebyshev's and Markov's Contributions......Page 156
4.1 Chebyshev's Moment Problem......Page 158
4.2.1 The Gaussian Procedure of Quadrature......Page 165
4.2.2 Generalizations of Gauss's Quadrature Formula, Systems of Orthogonal Polynomials......Page 169
4.2.3 Chebyshev's Contributions......Page 171
4.3.1 Markov's Early Work on Moments......Page 174
4.3.2 Stieltjes's Early Work on Moments......Page 177
4.4 Chebyshev's Further Work on Moments......Page 179
4.5 The Stieltjes Moment Problem......Page 184
4.6.1 Chebyshev's Probabilistic Work......Page 185
4.6.2 Chebyshev's Uncomplete Proof of the Central Limit Theorem from 1887......Page 188
4.6.3 Poincaré: Moments and Hypothesis of Elementary Errors......Page 191
4.6.4 Markov's Rigorous Proof......Page 192
4.7 Chebyshev's and Markov's Central Limit Theorem: Starting Point of a New Theory of Probability?......Page 200
4.7.2 Analytic Methods and Rigor......Page 202
4.7.3 The Role of the Central Limit Theorem in Chebyshev's and Markov's Work......Page 204
5 The Way Toward Modern Probability......Page 208
5.1.1 Lyapunov's Way Toward the Central Limit Theorem......Page 211
5.1.2 Nekrasov's Role in the Development of Probability Theory Around 1900......Page 212
5.1.3 Lyapunov Conditions and Lyapunov Inequality......Page 215
5.1.4 Sketch of Lyapunov's Proof for the Central Limit Theorem......Page 219
5.1.5 Markov's Reaction......Page 222
5.2.1 A New Generation......Page 225
5.2.2 Von Mises: Laplacian Method of Approximation, Complex and Real Adjunct......Page 228
5.2.3.1 Pólya's First Contributions......Page 235
5.2.3.2 The Hypothesis of Elementary Errors as a Motivation for Lévy's First Articles......Page 239
5.2.3.3 Poincaré and the Concept of Characteristic Functions......Page 241
5.2.3.4 Lévy's Fundamental Theorems on Characteristic Functions......Page 242
5.2.3.5 Pólya's Reaction to Lévy's First Articles......Page 246
5.2.4 Lindeberg: An Entirely New Method......Page 250
5.2.4.1 The Proof......Page 251
5.2.4.2 Different Theorems, Different Conditions......Page 253
5.2.5 Hausdorff's Reception of Lyapunov's, von Mises's, and Lindeberg's Work......Page 255
5.2.6.1 Stable Laws as Limit Laws......Page 259
5.2.6.2 The Functional Equation of the Characteristic Function of a Stable Law......Page 260
5.2.6.3 The Laws of Type L_{α,β}......Page 262
5.2.6.4 A Generalization of the Central Limit Theorem......Page 263
5.2.6.5 The "Classic'' Central Limit Theorem as a Special Case......Page 264
5.2.6.6 More Limit Laws......Page 266
5.2.6.7 Domains of Attraction of Stable Distributions......Page 267
5.2.7.1 The Statement......Page 270
5.2.7.2 The Proof......Page 273
5.2.8.1 Risk Theory as a Starting Point......Page 275
5.2.8.2 Cramér's Discussion of the Asymptotics of Edgeworth and Charlier A Expansions......Page 278
6.1 The Prehistory......Page 288
6.1.1 Lévy and the Problem of Un-negligible Summands......Page 289
6.1.2 Feller and the Case Which "does not belong to probability theory at all''......Page 292
6.2.1 Lévy's Main Theorems......Page 293
6.2.2 Lévy's "Intuitive'' Methods......Page 296
6.2.3.1 Lévy's Unproven Lemmata on Properties of Dispersion......Page 297
6.2.3.2 The "Classical Case''......Page 298
6.2.3.3 The "loi des grands nombres'' as a Sufficient Condition for the Central Limit Theorem......Page 300
6.2.3.4 Lévy's Decomposition Principle......Page 301
6.2.3.5 The "loi des grands nombres'' as a Necessary Condition in the Case of Identically Distributed Variables......Page 303
6.2.3.6 The "loi des grands nombres'' as a Necessary Condition in the General Case of Negligible Variables......Page 308
6.2.4 Feller's Theorems......Page 313
6.2.5.1 Auxiliary Theorems......Page 316
6.2.5.2 Main Theorem......Page 317
6.2.5.3 Criterion......Page 322
6.2.5.4 Necessity of Lindeberg Condition......Page 323
6.3 A Question of Priority?......Page 324
6.3.1 Lévy's and Feller's Results: A Comparison......Page 325
6.3.2 Another Question of Priority......Page 327
6.3.3 A Question of Methods and Style......Page 329
7.1.1 Measure-Theoretic Background......Page 332
7.1.2 Conditional Distribution and Expectation......Page 334
7.1.3 Lévy's Central Limit Theorem for Martingales......Page 336
7.2 Further Limit Problems......Page 342
7.2.1 Stochastic Processes with Independent Increments......Page 343
7.2.2 Limit Laws of Normed Sums......Page 346
7.3.1 Invariance Principles and Donsker's Theorem......Page 349
7.3.1.1 Wiener Measure and Wiener Integral......Page 350
7.3.1.2 Cameron and Martin......Page 353
7.3.1.3 The Invariance Principle......Page 355
7.3.1.4 Donsker's General Invariance Principle......Page 357
7.3.2 The Central Limit Theorem for Sums of Random Elements in Hilbert Spaces......Page 364
8 Conclusion: The Central Limit Theorem as a Link Between Classical and Modern Probability Theory......Page 370
References......Page 380
Name Index......Page 410
Subject Index......Page 416