Who is the Central Limit Theorem? Michael Schneier and Angela Jarrett

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…


1643 - Taj Mahal completed

1682 - Pennsylvania founded




1684 - Leibnitz calculus published

1687 - Newton principia published

James Bernoulli

1654-1705

Ars Conjectandi – 1713

published by Nicholas Bernoulli

Fundamental principles of calculus probabilities

Theorems of finite differences

Bernoulli trials

When are we?





When are we?





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?





1707 - England, Scotland and Wales form The United Kingdom of Great Britain






1707 - England, Scotland and Wales form The United Kingdom of Great Britain

1756 - 7 years’ war (French and Indian wars)


1765—James Watt invents steam engine

1770—Boston Massacre




1773—Boston Tea Party

1776—Declaration of Independence




1773—Boston Tea Party

1776—Declaration of Independence

1783—End of Revolutionary War, Treaty of Paris

1789—French Revolution


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 variables

Approximation formulae

Jean d’AlembertJoseph-Louis LagrangeAdrrien Marie Legendre

Carl Friedrich Gauss

When are we now?





When are we now?





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 work

The first attempt at estimating the error of the approximation (although unsucessful)

Peter Gustav Lejeune Dirichlet







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 theorem

The conditions he presented as necessary were considered too stringent.

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




• 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











1879 Thomas A. Edison invents practical electric light.

1893 New Zealand becomes first country in the world to grant women the vote.

Andrey Andreevich

Markov

1856-1922

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

1857-1928

Was also a student of Chebyshev

Developed what was considered the first real rigorous proof of the CLT using characteristic functions

Much 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?



1909—North Pole reportedly reached by American explorers Robert E. Peary and Matthew Henson.

1914—World War I begins

Historical context?



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 conditions

Served as a basis for future work

Feller + Levy

Feller + Levy

• The last significant work that was done on the CLT.

Feller + Levy


• Essentially “cleaned” up the work of Lindeberg

Feller + Levy



• Gave necessary and conditions that would go on to be proven sufficient later in the 20th century

Feller + Levy



• 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

Documents

Who is the Central Limit Theorem? Michael Schneier and Angela Jarrett