Who is the Central Limit Theorem?

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