Who is the Central Limit Theorem? Michael Schneier and Angela Jarrett.
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Transcript of Who is the Central Limit Theorem? Michael Schneier and Angela Jarrett.
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?
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
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
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 variables
Approximation formulae
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
• 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
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 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
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.
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
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?
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 conditions
Served as a basis for future work
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