A history of the central limit theorem: from classical to modern probability theory
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2011
|
| Schriftenreihe: | Sources and studies in the history of mathematics and physical sciences
|
| Schlagworte: | |
| Online-Zugang: | A history of the central limit theorem Inhaltsverzeichnis |
| Beschreibung: | XVI, 402 S. graph. Darst. |
| ISBN: | 9780387878560 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804145676318146560 |
|---|---|
| adam_text | Titel: A history of the central limit theorem
Autor: Fischer, Hans
Jahr: 2011
Contents
Preface............................................................ v
List of Figures ..................................................... xiii
Abbreviations and Denotations ...................................... xv
1 Introduction................................................... I
1.1 Different Versions of Central Limit Theorems................... 3
1.2 Objectives and Focus of the Present Examination................ 5
1.3 The Development of Analysis in the 19th Century............... 9
1.4 Literature on the History of the Central Limit Theorem........... II
1.5 Terminology and Notation................................... 12
1.6 The Prehistory: De Moivre s Theorem......................... 14
2 The Central Limit Theorem from Laplace to Cauchy:
Changes in Stochastic Objectives and in Analytical Methods........ 17
2.1 Laplace s Central Limit Theorem........................... 18
2.1.1 Sums of Independent Random Variables................. 19
2.1.2 Laplace s Method of Approximating Integrals,
and Algebraic Analysis ............................. 20
2.1.3 The Emergence of Characteristic Functions
and the Deduction of Approximating Normal Distributions . 21
2.1.4 The Rigor of Laplace s Analysis...................... 23
2.1.5 The Central Limit Theorem as a Tool of Good Sense ...... 25
2.1.5.1 The Comet Problem.......................... 25
2.1.5.2 The Foundation of the Method of Least Squares .. 26
2.1.5.3 Benefits from Games of Chance................ 30
2.2 Poisson s Modifications..................................... 31
2.2.1 Poisson s Concept of Random Variable.................. 31
2.2.2 Poisson s Representation of the Probabilities of Sums...... 32
Contents
2.2.3 The Role of the Central Limit Theorem in Poisson s Work .. 33
2.2.3. t Poisson s Version of the Central Limit Theorem .. 33
2.2.3.2 Poisson s Law of Large Numbers............... 35
2.2.4 Poisson s Infinitistic Approach......................... 36
2.2.5 Approximation by Series Expansions ................... 39
2.3 The Central Limit Theorem After Poisson...................... 40
2.3.1 Toward a New Conception of Mathematics............... 40
2.3.2 Changes in the Status of Probability Theory.............. 42
2.3.3 The Rigorization of Laplace s Idea of Approximation...... 42
2.4 Dirichlet s Proof of the Central Limit Theorem.................. 44
2.4.1 Dirichlet s Modification of the Laplacian Method
of Approximation.................................... 44
2.4.2 The Application of the Discontinuity Factor.............. 46
2.4.3 Dirichlet s Proof..................................... 47
2.4.3.1 Tacit Assumptions and Proposition ............. 48
2.4.3.2 Dirichlet s Discussion of the Limit.............. 48
2.5 Cauchy s Bound for the Error of Approximation................. 52
2.5.1 The Cauchy-Bienaymé Dispute........................ 52
2.5.2 Cauchy s Exceptional Laws of Error.................... 55
2.5.3 Bienaymé s Arguments............................... 59
2.5.4 Cauchy s Version of the Central Limit Theorem........... 61
2.5.5 Cauchy s Idea of Proof............................... 63
2.5.6 The End of the Controversy ........................... 65
2.5.7 Conclusion: Steps Toward Modern Probability............ 67
Appendix: Original Text of Dirichlet s Proof of the Central Limit
Theorem According to Lecture Notes from 1846................ 69
The Hypothesis of Elementary Errors............................ 75
3.1 Gauss and His Error Law.................................. 76
3.2 Hagen, Bessel, and elementare Fehler ........................ 79
3.2.1 The Rediscovery of the Hypothesis of Elementary Errors
by Gotthilf Hagen.................................... 80
3.2.2 Bessel s Generalization of the Hypothesis of Elementary
Errors.............................................. 87
3.3 The Reception of Hagen s and Bessel s Ideas................... 93
3.3.1 Normal Distributions in Statistics of Biological
and Social Phenomena................................ 93
3.3.2 Advancement Within Error Theory ..................... 95
3.3.2.1 Rectangularly Distributed Elementary Errors..... 96
3.3.2.2 Crofton s Hypothesis......................... 98
3.3.2.3 Pizzetti s Account on the Hypothesis
of Elementary Errors......................... 102
3.3.2.4 Schols, and Elementary Errors in Plane and Space. 104
Contents
3.4 Nonnormal Distributions, Series Expansions, and Modifications
of the Hypothesis of Elementary Errors........................107
3.4.1 Approximations of Arbitrary Probability Functions
by Series in Hermite Polynomials ......................109
3.4.2 The Natural Role of the Normal Distribution
and Its Derivatives...................................115
3.4.2.1 Hausdorff s Kanonische Parameter ...........116
3.4.2.2 Charlier s A Series...........................119
3.4.2.3 Edgeworth and The Law of Error.............122
3.4.3 The Method of Translation............................132
3.4.3.1 The Log-Normal Distribution..................133
3.4.3.2 Wicksell s General Model of Elementary Errors ..135
3.4.3.3 The Further Fate of the Hypothesis of Elementary
Errors......................................136
Appendix: Letter from Bessel to Jacobi, 14 August 1834..............138
Chebyshev s and Markov s Contributions.........................139
4.1 Chebyshev s Moment Problem...............................141
4.2 Quadrature Formulae, Continued Fractions, Orthogonal
Polynomials, Moments......................................148
4.2.1 The Gaussian Procedure of Quadrature..................148
4.2.2 Generalizations of Gauss s Quadrature Formula, Systems
of Orthogonal Polynomials............................152
4.2.3 Chebyshev s Contributions............................154
4.3 Moment Problems Around 1884: Markov and Stieltjes ...........157
4.3.1 Markov s Early Work on Moments......................157
4.3.2 Stieltjes s Early Work on Moments.....................160
4.4 Chebyshev s Further Work on Moments........................162
4.5 The Stieltjes Moment Problem ...............................167
4.6 Moment Theory and Central Limit Theorem....................168
4.6.1 Chebyshev s Probabilistic Work........................168
4.6.2 Chebyshev s Uncomplete Proof of the Central
Limit Theorem from 1887.............................171
4.6.3 Poincaré: Moments and Hypothesis of Elementary Errors ..174
4.6.4 Markov s Rigorous Proof.............................175
4.7 Chebyshev s and Markov s Central Limit Theorem: Starting Point
of a New Theory of Probability?..............................183
4.7.1 Random Variables and Limit Theorems..................185
4.7.2 Analytic Methods and Rigor...........................185
4.7.3 The Role of the Central Limit Theorem in Chebyshev s
and Markov s Work..................................187
Contents
The Way Toward Modern Probability............................191
5.1 Russian Contributions Between the Turn of the Century
and the First World War.....................................194
5.1.1 Lyapunov s Way Toward the Central Limit Theorem.......194
5.1.2 Nekrasov s Role in the Development of Probability
Theory Around 1900.................................195
5.1.3 Lyapunov Conditions and Lyapunov Inequality...........198
5.1.4 Sketch of Lyapunov s Proof for the Central Limit Theorem . 202
5.1.5 Markov s Reaction...................................205
5.2 The Central Limit Theorem in the Twenties.....................208
5.2.1 A New Generation...................................208
5.2.2 Von Mises: Laplacian Method of Approximation,
Complex and Real Adjunct............................211
5.2.3 Pólya and Levy: Laws of Error, Moments
and Characteristic Functions...........................218
5.2.3.1 Pólya s First Contributions....................218
5.2.3.2 The Hypothesis of Elementary Errors
as a Motivation for Levy s First Articles.........222
5.2.3.3 Poincaré and the Concept of Characteristic
Functions...................................224
5.2.3.4 Levy s Fundamental Theorems on Characteristic
Functions...................................225
5.2.3.5 Pólya s Reaction to Levy s First Articles.........229
5.2.4 Lindeberg: An Entirely New Method....................233
5.2.4.1 The Proof..................................234
5.2.4.2 Different Theorems, Different Conditions........236
5.2.5 Hausdorff s Reception of Lyapunov s, von Mises s,
and Lindeberg s Work................................238
5.2.6 Levy s Discussion of Stable Laws in His Calcul des
probabilités.........................................242
5.2.6.1 Stable Laws as Limit Laws....................242
5.2.6.2 The Functional Equation of the Characteristic
Function of a Stable Law......................243
5.2.6.3 The Laws of Type La ß.......................245
5.2.6.4 A Generalization of the Central Limit Theorem ... 246
5.2.6.5 The Classic Central Limit Theorem
as a Special Case ............................247
5.2.6.6 More Limit Laws............................249
5.2.6.7 Domains of Attraction of Stable Distributions ....250
5.2.7 Bernshtein and His lemme fondamental ................253
5.2.7.1 The Statement...............................253
5.2.7.2 The Proof..................................256
Contents
5.2.8 Cramer: Lyapunov Bounds and Asymptotic Behavior
of Exponential Series ...............................258
5.2.8.1 Risk Theory as a Starting Point ................258
5.2.8.2 Cramer s Discussion of the Asymptotics of
Edgeworth and Charlier A Expansions ..........261
Levy and Feller on Normal Limit Distributions around 1935........271
6.1 The Prehistory.............................................271
6.1.1 Levy and the Problem of Un-negligible Summands........272
6.1.2 Feller and the Case Which does not belong to probability
theory at all .......................................275
6.2 Levy s and Feller s Results and Methods.......................276
6.2.1 Levy s Main Theorems...............................276
6.2.2 Levy s Intuitive Methods............................279
6.2.3 Levy s Proofs.......................................280
6.2.3.1 Levy s Unproven Lemmata on Properties
of Dispersion................................280
6.2.3.2 The Classical Case .........................281
6.2.3.3 The loi des grands nombres as a Sufficient
Condition for the Central Limit Theorem........283
6.2.3.4 Levy s Decomposition Principle................284
6.2.3.5 The loi des grands nombres as a Necessary
Condition in the Case of Identically Distributed
Variables...................................286
6.2.3.6 The loi des grands nombres as a Necessary
Condition in the General Case of Negligible
Variables...................................291
6.2.4 Feller s Theorems....................................296
6.2.5 Feller s Proofs.......................................299
6.2.5.1 Auxiliary Theorems..........................299
6.2.5.2 Main Theorem ..............................300
6.2.5.3 Criterion ...................................305
6.2.5.4 Necessity of Lindeberg Condition..............306
6.3 A Question of Priority?......................................307
6.3.1 Levy s and Feller s Results: A Comparison...............308
6.3.2 Another Question of Priority...........................310
6.3.3 A Question of Methods and Style.......................312
Generalizations................................................315
7.1 Levy on Sums of Nonindependent Random Variables............315
7.1.1 Measure-Theoretic Background........................315
7.1.2 Conditional Distribution and Expectation................317
7.1.3 Levy s Central Limit Theorem for Martingales ...........319
7.2 Further Limit Problems .....................................325
7.2.1 Stochastic Processes with Independent Increments........326
7.2.2 Limit Laws of Normed Sums..........................329
xii Contents
7.3 Extensions of the Central Limit Theorem to Stochastic Processes
and Random Elements in Metric Spaces........................332
7.3.1 Invariance Principles and Donsker s Theorem ............332
7.3.1.1 Wiener Measure and Wiener Integral............333
7.3.1.2 Cameron and Martin .........................336
7.3.1.3 The Invariance Principle......................338
7.3.1.4 Donsker s General Invariance Principle..........340
7.3.2 The Central Limit Theorem for Sums of Random Elements
in Hubert Spaces ....................................347
8 Conclusion: The Central Limit Theorem as a Link
Between Classical and Modern Probability Theory.................353
References.........................................................363
Name Index........................................................393
Subject Index......................................................399
|
| any_adam_object | 1 |
| author | Fischer, Hans 1954- |
| author_GND | (DE-588)140046151 |
| author_facet | Fischer, Hans 1954- |
| author_role | aut |
| author_sort | Fischer, Hans 1954- |
| author_variant | h f hf |
| building | Verbundindex |
| bvnumber | BV037388205 |
| classification_rvk | SK 800 |
| ctrlnum | (OCoLC)700628283 (DE-599)DNB990120546 |
| discipline | Mathematik |
| era | Geschichte 1810-1950 gnd |
| era_facet | Geschichte 1810-1950 |
| format | Book |
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| genre | 1\p (DE-588)4113937-9 Hochschulschrift gnd-content |
| genre_facet | Hochschulschrift |
| id | DE-604.BV037388205 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:23:14Z |
| institution | BVB |
| isbn | 9780387878560 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022541155 |
| oclc_num | 700628283 |
| open_access_boolean | |
| owner | DE-824 DE-12 DE-83 DE-945 |
| owner_facet | DE-824 DE-12 DE-83 DE-945 |
| physical | XVI, 402 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Sources and studies in the history of mathematics and physical sciences |
| spelling | Fischer, Hans 1954- Verfasser (DE-588)140046151 aut Die verschiedenen Formen und Funktionen des zentralen Grenzwertsatzes in der Entwicklung von der klassischen bis zur modernen Wahrscheinlichkeitsrechnung A history of the central limit theorem from classical to modern probability theory Hans Fischer New York, NY [u.a.] Springer 2011 XVI, 402 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Sources and studies in the history of mathematics and physical sciences Geschichte 1810-1950 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Zentraler Grenzwertsatz (DE-588)4067618-3 gnd rswk-swf 1\p (DE-588)4113937-9 Hochschulschrift gnd-content Zentraler Grenzwertsatz (DE-588)4067618-3 s Geschichte 1810-1950 z DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Geschichte (DE-588)4020517-4 s 2\p DE-604 3\p DE-604 Erscheint auch als Online-Ausgabe 978-0-387-87857-7 http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=4061620&custom_att_2=simple_viewer Verlagsdaten Springer A history of the central limit theorem HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022541155&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Fischer, Hans 1954- A history of the central limit theorem from classical to modern probability theory Geschichte (DE-588)4020517-4 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Zentraler Grenzwertsatz (DE-588)4067618-3 gnd |
| subject_GND | (DE-588)4020517-4 (DE-588)4064324-4 (DE-588)4067618-3 (DE-588)4113937-9 |
| title | A history of the central limit theorem from classical to modern probability theory |
| title_alt | Die verschiedenen Formen und Funktionen des zentralen Grenzwertsatzes in der Entwicklung von der klassischen bis zur modernen Wahrscheinlichkeitsrechnung |
| title_auth | A history of the central limit theorem from classical to modern probability theory |
| title_exact_search | A history of the central limit theorem from classical to modern probability theory |
| title_full | A history of the central limit theorem from classical to modern probability theory Hans Fischer |
| title_fullStr | A history of the central limit theorem from classical to modern probability theory Hans Fischer |
| title_full_unstemmed | A history of the central limit theorem from classical to modern probability theory Hans Fischer |
| title_short | A history of the central limit theorem |
| title_sort | a history of the central limit theorem from classical to modern probability theory |
| title_sub | from classical to modern probability theory |
| topic | Geschichte (DE-588)4020517-4 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Zentraler Grenzwertsatz (DE-588)4067618-3 gnd |
| topic_facet | Geschichte Wahrscheinlichkeitsrechnung Zentraler Grenzwertsatz Hochschulschrift |
| url | http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=4061620&custom_att_2=simple_viewer http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022541155&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT fischerhans dieverschiedenenformenundfunktionendeszentralengrenzwertsatzesinderentwicklungvonderklassischenbiszurmodernenwahrscheinlichkeitsrechnung AT fischerhans ahistoryofthecentrallimittheoremfromclassicaltomodernprobabilitytheory |