Who is the Central Limit Theorem? Michael Schneier and Angela Jarrett
Feb 22, 2016
Who is theCentral Limit
Theorem?Michael Schneier and Angela Jarrett
Figuratively…
Figuratively…
“Frankenstein’s Monster”
Framework• When talking about the history of the Central
Limit Theorem (CLT) need to define what we are talking about
Framework• When talking about the history of the Central
Limit Theorem (CLT) need to define what we are talking about
• Important to understand that the present day CLT looks nothing like what mathematicians were working on in the 18th century
Travel back in time…
Travel back in time… 1643 - Taj Mahal completed 1682 - Pennsylvania founded
Travel back in time… 1643 - Taj Mahal completed 1682 - Pennsylvania founded 1684 - Leibnitz calculus published 1687 - Newton principia published
James Bernoulli1654-1705
Ars Conjectandi – 1713published by Nicholas Bernoulli
Fundamental principles of calculus probabilitiesTheorems of finite differences Bernoulli trials
When are we? 1643 - Taj Mahal completed 1682 - Pennsylvania founded 1684 - Leibnitz calculus published 1687 - Newton principia published
When are we? 1643 - Taj Mahal completed 1682 - Pennsylvania founded 1684 - Leibnitz calculus published 1687 - Newton principia published 1707 - England,Scotland and Wales form The
United Kingdom of Great Britain
Abraham de Moivre
1667-1754
Doctrine of Chances—1718
Approximating the sum of the terms of a binomial expansion using approximation for factorials
Gottfried Wilhelm LeibnitzIsaac Newton
Pierre Rémond de MontmortJohn BernoulliJames Stirling
What’s been going on?
1643 - Taj Mahal completed 1682 - Pennsylvania founded 1684 - Leibnitz calculus published 1687 - Newton principia published 1707 - England, Scotland and Wales form The
United Kingdom of Great Britain
What’s been going on?
1643 - Taj Mahal completed 1682 - Pennsylvania founded 1684 - Leibnitz calculus published 1687 - Newton principia published 1707 - England, Scotland and Wales form The
United Kingdom of Great Britain 1756 - 7 years’ war (French and Indian wars)
What’s been going on?
1765—James Watt invents steam engine 1770—Boston Massacre
What’s been going on?
1765—James Watt invents steam engine 1770—Boston Massacre 1773—Boston Tea Party 1776—Declaration of Independence
What’s been going on?
1765—James Watt invents steam engine 1770—Boston Massacre 1773—Boston Tea Party 1776—Declaration of Independence 1783—End of Revolutionary War, Treaty of Paris 1789—French Revolution
What’s been going on?
1793—Louis XVI and Marie Antoinette executed (Reign of Terror begins in France)
1799—Rosetta Stone found
What’s been going on?
1793—Louis XVI and Marie Antoinette executed (Reign of Terror begins in France)
1799—Rosetta Stone found 1803—Louisiana Purchase 1807—First successful steamboat trip
Pierre-Simon Laplace
1749-1827
Memoire sur les approximations des formules qui sont fonctions des tres grands nombres et sur leur application aux probabilites—1810
Theorie analytique des probabilites—1812
Sums of independent random variablesApproximation formulae
Jean d’AlembertJoseph-Louis LagrangeAdrrien Marie Legendre
Carl Friedrich Gauss
When are we now? 1793—Louis XVI and Marie Antoinette executed
(Reign of Terror begins in France) 1799—Rosetta Stone found 1803—Louisiana Purchase 1807—First successful steamboat trip
When are we now? 1793—Louis XVI and Marie Antoinette executed
(Reign of Terror begins in France) 1799—Rosetta Stone found 1803—Louisiana Purchase 1807—First successful steamboat trip 1825—First passenger carrying railroad 1833—Slavery abolished in British empire
Siméon-Denis Poisson
1781-1840
Recherches sur la probabilite des jugements—1837
Law of Large Numbers
Turning point toward mathematical rigor for stochastics
CLT after Poisson
CLT after Poisson• Following Poisson probability theory and
mathematics as a whole underwent significant changes
CLT after Poisson• Following Poisson probability theory and
mathematics as a whole underwent significant changes
• Mathematics was no longer viewed as having to have an association with the sciences
1805-1859
Managed to prove a signifcant amount of Poissons workThe first attempt at estimating the error of the approximation (although unsucessful)
Peter Gustav Lejeune Dirichlet
What’s been going on?
1825—First passenger carrying railroad 1833—Slavery abolished in British empire
What’s been going on?
1825—First passenger carrying railroad 1833—Slavery abolished in British empire 1851—Herman Melville's Moby-Dick.
1861—U.S. Civil War begins
Augustin-Louis Cauchy
1789-1857
Cauchy managed to prove the convergence of the CLT theoremThe conditions he presented as necessary were considered too stringent.
CLT in the 20th century
CLT in the 20th century• There were three majors problems with the
work done with Dirchlet and Cauchy
CLT in the 20th century• There were three majors problems with the
work done with Dirchlet and Cauchy• 1. The theorem was not proved for distributions
with infinite support
CLT in the 20th century• There were three majors problems with the
work done with Dirchlet and Cauchy• 1. The theorem was not proved for distributions
with infinite support• 2. There were no explicit conditions, in terms of
the moments, under which the theorem would hold
CLT in the 20th century• There were three majors problems with the
work done with Dirchlet and Cauchy• 1. The theorem was not proved for distributions
with infinite support• 2. There were no explicit conditions, in terms of
the moments, under which the theorem would hold
• 3. The rate of convergence for the theorem was not studied.
Pafnuty Chebyshev
1821-1894
What’s been going on?
1825—First passenger carrying railroad 1833—Slavery abolished in British empire 1851—Herman Melville's Moby-Dick.
1861—U.S. Civil War begins
What’s been going on?
1825—First passenger carrying railroad 1833—Slavery abolished in British empire 1851—Herman Melville's Moby-Dick.
1861—U.S. Civil War begins 1879 Thomas A. Edison invents practical
electric light. 1893 New Zealand becomes first country in the
world to grant women the vote.
Andrey Andreevich
Markov1856-1922
Chebyshev and Markov provided the first truly rigorous proof of the CLT
They did so using the method of moments developed by Chebyshev
Aleksandr Mikhailovich Lyapunov
1857-1928Was also a student of ChebyshevDeveloped what was considered the first real rigorous proof of the CLT using characteristic functionsMuch of the work on the CLT in the later half of the 20th century would be based on this
Historical context?
Historical context? 1895—X-rays discovered by German physicist
Wilhelm Roentgen 1903—Wright brothers, Orville and Wilbur, fly
first powered, controlled, heavier-than-air plane at Kitty Hawk, N.C.
Historical context? 1895—X-rays discovered by German physicist
Wilhelm Roentgen 1903—Wright brothers, Orville and Wilbur, fly
first powered, controlled, heavier-than-air plane at Kitty Hawk, N.C.
1909—North Pole reportedly reached by American explorers Robert E. Peary and Matthew Henson.
1914—World War I begins
Historical context? 1895—X-rays discovered by German physicist
Wilhelm Roentgen 1903—Wright brothers, Orville and Wilbur, fly first
powered, controlled, heavier-than-air plane at Kitty Hawk, N.C.
1909—North Pole reportedly reached by American explorers Robert E. Peary and Matthew Henson.
1914—World War I begins 1920—League of Nations holds first meeting at
Geneva.
Jarl Waldemar Lindeberg
1876-1932
Published an elementary proof the CLT with a few distinct advantages• 1. It can be applied in a very
general context • 2. It takes the rate of
convergence in the limit theorem under consideration.
Only provided necessary conditionsServed as a basis for future work
Feller + Levy
Feller + Levy• The last significant work that was done on the
CLT.
Feller + Levy• The last significant work that was done on the
CLT.• Essentially “cleaned” up the work of Lindeberg
Feller + Levy• The last significant work that was done on the
CLT.• Essentially “cleaned” up the work of Lindeberg• Gave necessary and conditions that would go on
to be proven sufficient later in the 20th century
Feller + Levy• The last significant work that was done on the
CLT.• Essentially “cleaned” up the work of Lindeberg• Gave necessary and conditions that would go on
to be proven sufficient later in the 20th century• Their versions of the CLT are essentially what we
use today.
References http://www.mhhe.com/math/calc/smithminton2e/cd/tools/timeline/bernoulli.html http://www.maths.tcd.ie/pub/HistMath/People/Bernoullis/RouseBall/RB_Bernoullis.html http://www.britannica.com/EBchecked/topic/62599/Jakob-Bernoulli http://www.infoplease.com/ipa/A0001196.html H. Fischer, A History of the Central Limit Theorem, Sources and Studies in the History of
Mathematics and Physical Sciences, DOI 10.1007/978-0-387-87857-7_2, http://www.cs.xu.edu/math/Sources/Laplace/index.html.
http://wiki.math.toronto.edu/TorontoMathWiki/images/0/00/MAT1000DanielRuedt.pdf Games, Gods and Gambling, F. N. David, 1962, Hafner Publishing Company, New York Fundamental Limit Theorems of Probability Theory, M. Loéve, Annals of Mathematical
Statistics, pp. 321-338, 1950 Book Reviews / Historia Mathematica 39 (2012) 460–479 , Menso Folkerts , University of
Munich, http://dx.doi.org/10.1016/j.hm.2012.03.006
Image Sources http://www.quien.net/jakob-bernoulli.php
http://www.famous-mathematicians.com/abraham-de-moivre/
http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz
http://www.crystalinks.com/newton.html
http://en.wikipedia.org/wiki/Johann_Bernoulli
http://portrait.kaar.at/Naturwissenschaftler/image45.html
http://es.wikipedia.org/wiki/Jean_le_Rond_d'Alembert
http://en.wikipedia.org/wiki/Joseph-Louis_Lagrange
http://en.wikipedia.org/wiki/Adrien-Marie_Legendre
http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
http://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson
http://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet
http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy
http://en.wikipedia.org/wiki/Pafnuty_Chebyshev
http://logic.pdmi.ras.ru/Markov/
http://en.wikipedia.org/wiki/Aleksandr_Lyapunov
http://www.geni.com/people/Jarl-Waldemar-Lindeberg/5277851302290074410