Who is the Central Limit Theorem?
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Transcript Who is the Central Limit Theorem?
Who is the
Central 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 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
Gottfried Wilhelm Leibnitz
Isaac Newton
Pierre Rémond de Montmort
John Bernoulli
James 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
Mémoire sur les approximations des
formules qui sont fonctions des très grands
nombres et sur leur application aux
probabilités—1810
Théorie analytique des probabilités—
1812
Sums of independent random
variables
Approximation formulae
Jean d’Alembert
Joseph-Louis Lagrange
Adrrien 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 probabilité 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
Peter Gustav
Lejeune Dirichlet
1805-1859
Managed to prove a signifcant
amount of Poissons work
The first attempt at estimating
the error of the approximation
(although unsucessful)
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
th
20
century
CLT in the
•
th
20
century
There were three majors problems with the work done
with Dirchlet and Cauchy
CLT in the
th
20
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
th
20
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
th
20
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
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?
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
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://en.wikipedia.org/wiki/Adrien-Marie_Legendre
http://www.famous-mathematicians.com/abraham-demoivre/
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_D
irichlet
http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy
http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leib
niz
http://www.crystalinks.com/newton.html
http://en.wikipedia.org/wiki/Johann_Bernoulli
http://portrait.kaar.at/Naturwissenschaftler/image45.h
tml
http://en.wikipedia.org/wiki/Pafnuty_Chebyshev
http://es.wikipedia.org/wiki/Jean_le_Rond_d'Alember
t
http://logic.pdmi.ras.ru/Markov/
http://en.wikipedia.org/wiki/Joseph-Louis_Lagrange
http://www.geni.com/people/Jarl-WaldemarLindeberg/5277851302290074410
http://en.wikipedia.org/wiki/Aleksandr_Lyapunov