Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Tercentennial anniversary of Bernoulli’s law of large numbers
HTML articles powered by AMS MathViewer

by Manfred Denker PDF
Bull. Amer. Math. Soc. 50 (2013), 373-390 Request permission
References
  • Jon Aaronson and Manfred Denker, The Poincaré series of $\mathbf C\sbs \mathbf Z$, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 1–20. MR 1676950, DOI 10.1017/S0143385799126592
  • Anatole Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), 329–334. MR 133857, DOI 10.1090/S0002-9939-1962-0133857-9
  • V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms, Ann. of Math. (2) 150 (1999), no. 3, 977–1027. MR 1740988, DOI 10.2307/121060
  • J. Bernoulli, Ars conjectandi: Opus posthumum: accedit tractatus de seriebus infinitis, et Epistola Gallicè Scripta de ludo pilae reticularis. Published 1713 by Impensis Thurnisiorum, fratrum in Basileae. (Latin.)
  • J. Bernoulli: Wahrscheinlichkeitsrechnung: Ars conjectandi. 1.,2., und 4. Theil (1913). Edited and translated by R. Haussner. Oswalds Klassiker der exakten Wissenschaften Band 107/108. Reprint. Verlag Harri Deutsch, 1999. (German.)
  • Jacob Bernoulli, The art of conjecturing, Johns Hopkins University Press, Baltimore, MD, 2006. Together with “Letter to a friend on sets in court tennis”; Translated from the Latin and with an introduction and notes by Edith Dudley Sylla. MR 2195221
  • J. Bernoulli, On the law of large numbers. Translated to English by Oscar Sheynin, Berlin 2005. http://www.sheynin.de/download/bernoulli.pdf.
  • N. Bernoulli, De usu artis conjectandi in jure. Basel 1709. Reprint in: Die Werke von Jakob Bernoulli, Vol. 3, 287–326, Birkhäuser Basel 1975. (Latin.)
  • Andrew C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122–136. MR 3498, DOI 10.1090/S0002-9947-1941-0003498-3
  • R. N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1986. Reprint of the 1976 original. MR 855460
  • Patrick Billingsley, The Lindeberg-Lévy theorem for martingales, Proc. Amer. Math. Soc. 12 (1961), 788–792. MR 126871, DOI 10.1090/S0002-9939-1961-0126871-X
  • Gunnar A. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 3, 561–574. MR 957261, DOI 10.1017/S0305004100065750
  • H. Cramér, Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques et Industrielles 736 (1938), 5–23. Colloque consacré à la théorie de probabilités. Vol. 3, Hermann, Paris. (French.)
  • S. G. Dani and M. McCrudden, Embeddability of infinitely divisible distributions on linear Lie groups, Invent. Math. 110 (1992), no. 2, 237–261. MR 1185583, DOI 10.1007/BF01231332
  • S. G. Dani and M. McCrudden, Convolution roots and embeddings of probability measures on Lie groups, Adv. Math. 209 (2007), no. 1, 198–211. MR 2294221, DOI 10.1016/j.aim.2006.05.002
  • Paul Deheuvels, Luc Devroye, and James Lynch, Exact convergence rate in the limit theorems of Erdős-Rényi and Shepp, Ann. Probab. 14 (1986), no. 1, 209–223. MR 815966
  • A. de Moivre, The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play. London 1718. Second edition, London 1738, third edition, London 1756.
  • Manfred Denker and Susanne Koch, Almost sure local limit theorems, Statist. Neerlandica 56 (2002), no. 2, 143–151. Special issue: Frontier research in theoretical statistics, 2000 (Eindhoven). MR 1916315, DOI 10.1111/1467-9574.00189
  • J.-D. Deuschel, D. W. Strook, Large Deviations. AMS Chelsea Publ., Amer. Math. Soc., Providence 1989.
  • Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1–125. MR 14626, DOI 10.1007/BF02392223
  • N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 1, 119–122. MR 606010, DOI 10.1007/BF01013465
  • W. Feller, Über das Gesetz der großen Zahlen. Acta Litt. Scient. Szeged 8 (1937), 191–201. (German.)
  • James Franklin, The science of conjecture, Johns Hopkins University Press, Baltimore, MD, 2001. Evidence and probability before Pascal. MR 1893301
  • C. F. Gauß, Theoria combinationis observationum erroribus minimis obnoxiae. Commentationes Societatis Regiae Scientiarum Gottingensis recentiores 5 (classis mathematicae) 1823. Part 1: February 15, 1821, Part 2: February 2, 1823. (Theory of combinations of observations which are subject to small errors.) Publ. H. Dieterich 1825. (Latin.)
  • B. V. Gnedenko, O lokal’noĭ predel’noĭ theoreme teorii veroyatnosteĭ. Uspehi matemat. nauk 3 (1948), 187–194. (On the local limit theorem in the theory of probability.) (Russian)
  • B. V. Gnedenko, O lokal’noĭ predel’noĭ teoreme dlya odinakovo raspredelennyh nezavisimyh slagaemyh. Wiss. Z. Humboldt–Univ. Berlin. Math.-Naturwiss. Reihe 3 (1953/4), 287–293. (On the local limit theorem for identically distributed independent terms.)
  • Anders Hald, A history of probability and statistics and their applications before 1750, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR 1029276, DOI 10.1002/0471725161
  • I. A. Ibragimov, A central limit theorem for a class of dependent random variables, Teor. Verojatnost. i Primenen. 8 (1963), 89–94 (Russian, with English summary). MR 0151997
  • A. Khintchine, Über einen neuen Grenzwertsatz der Wahrscheinlichkeitsrechnung, Math. Ann. 101 (1929), no. 1, 745–752 (German). MR 1512565, DOI 10.1007/BF01454873
  • A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1977 (German). Reprint of the 1933 original. MR 0494348
  • A. Kolmogoroff, Über die Summen durch den Zufall bestimmter unabhängiger Größen. Math. Ann. 99 (1928), 309–319; Math. Ann. 102 (1929), 484–488.
  • Ekkehard Krätzel and Werner Georg Nowak, Lattice points in large convex bodies, Monatsh. Math. 112 (1991), no. 1, 61–72. MR 1122105, DOI 10.1007/BF01321717
  • Michael T. Lacey and Walter Philipp, A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), no. 3, 201–205. MR 1045184, DOI 10.1016/0167-7152(90)90056-D
  • E. Landau, Zur analytischen Zahlentheorie der definiten quadratischen Formen (Über die Gitterpunkte in einem mehrdimensionalen Ellipsoid). Sitzungsberichte Preuss. Akad. Wiss. 31 (1915), 458–476. (German.)
  • A. Liapounoff, Nouvelle forme du théorème sur la limite de probabilité. Mémoires de l’Académie Impériale des Sciences de St.-Pétersbourg VIII$^\circ$ série, 12(5) (1901), 1–24. (French.)
  • G. A. Margulis, Discrete subgroups and ergodic theory, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 377–398. MR 993328
  • Michael B. Marcus and Wojbor A. Woyczyński, Stable measures and central limit theorems in spaces of stable type, Trans. Amer. Math. Soc. 251 (1979), 71–102. MR 531970, DOI 10.1090/S0002-9947-1979-0531970-2
  • Kh. O. Ondar, ed., The Correspondence Between A.A. Markov and A.A. Chuprov on the Theory of Probability and Mathematical Statistics. Translated from the Russian O teorii veroiatnosteĭ i matematicheskoĭ statistike by Charles and Margaret Stein. Springer Verlag, New York, Heidelberg, Berlin 1981.
  • K. Pearson, James Bernoulli’s theorem. Biometrika 17 (1925), 201–210.
  • V. V. Petrov, O lokal’nyh predel’nyh teoremah dlya summ nezavisimyh slučaĭnyh veličin. Teoriya veroyatn. i ee primen. 9 No. 2 (1964), 343–352. On local limit theorems for sums of independent random variables. Theor. Probab. Appl. 9 No. 2 (1964), 312–320.
  • V. V. Petrov, Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. MR 0388499
  • Ivo Schneider, Direct and indirect influences of Jakob Bernoulli’s Ars conjectandi in 18th century Great Britain, J. Électron. Hist. Probab. Stat. 2 (2006), no. 1, 17 (English, with English and French summaries). MR 2393219
  • G. Shafer, The significance of Jacob Bernouilli’s Ars Conjectandi for the philosophy of probability today. J. of Econometrics 75 (1996), 15–32.
  • L. A. Shepp, A local limit theorem, Ann. Math. Statist. 35 (1964), 419–423. MR 166817, DOI 10.1214/aoms/1177703766
  • S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math. 19 (1966), 261–286. MR 203230, DOI 10.1002/cpa.3160190303
  • B. L. van der Waerden, Kommentar zu den Meditationes und der Ars Conjectandi. In: Die Werke von Jakob Bernoulli, Vol. 3, 353–383, Birkhäuser Basel 1975. (German.)
  • R. v. Mises, Grundlagen der Wahrscheinlichkeitsrechnung, Math. Z. 5 (1919), no. 1-2, 52–99 (German). MR 1544374, DOI 10.1007/BF01203155
Similar Articles
Additional Information
  • Manfred Denker
  • Affiliation: Department of Mathematics, The Pennsylvania State University, State College, Pennsylvania 16802
  • Email: denker@math.psu.edu
  • Published electronically: March 28, 2013
  • Additional Notes: The author would like to thank Brian Nowakowski and an anonymous referee for valuable comments which led to several improvements of the text.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 50 (2013), 373-390
  • MSC (2010): Primary 60-03; Secondary 01-01, 60F05, 62-03
  • DOI: https://doi.org/10.1090/S0273-0979-2013-01411-3
  • MathSciNet review: 3049869