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| {{Dablink|This article is about the method of moments in [[probability theory]]. See [[method of moments]] for other techniques bearing the same name.}}
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| In [[probability theory]], the '''method of moments''' is a way of proving [[convergence in distribution]] by proving convergence of a sequence of [[moment (mathematics)|moment]] sequences.<ref>{{cite book|last=Prokhorov|first=A.V.|chapter=Moments, method of (in probability theory)|title=Encyclopaedia of Mathematics (online)|isbn=1-4020-0609-8|url=http://eom.springer.de/m/m064610.htm|mr=1375697|editor=M. Hazewinkel}}</ref> Suppose ''X'' is a [[random variable]] and that all of the moments
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| :<math>\operatorname{E}(X^k)\,</math>
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| exist. Further suppose the [[probability distribution]] of ''X'' is completely determined by its moments, i.e., there is no other probability distribution with the same sequence of moments
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| (cf. the [[problem of moments]]). If
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| :<math>\lim_{n\to\infty}\operatorname{E}(X_n^k) = \operatorname{E}(X^k)\,</math>
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| for all values of ''k'', then the sequence {''X''<sub>''n''</sub>} converges to ''X'' in distribution.
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| The method of moments was introduced by [[Pafnuty Chebyshev]] for proving the [[central limit theorem]]; Chebyshev cited earlier contributions by [[Irénée-Jules Bienaymé]].<ref>{{cite book|mr=2743162|last=Fischer|first=H.|title=A history of the central limit theorem. From classical to modern probability theory.|series= Sources and Studies in the History of Mathematics and Physical Sciences|publisher=Springer|location=New York|year=2011|isbn=978-0-387-87856-0|chapter=4. Chebyshev's and Markov's Contributions.}}</ref> More recently, it has been applied by [[Eugene Wigner]] to prove [[Wigner's semicircle law]], and has since found numerous applications in the [[random matrix theory|theory of random matrices]].<ref>{{cite book|last=Anderson|first=G.W.|last2=Guionnet|first2=A.|last3=Zeitouni|first3=O.|title=An introduction to random matrices.|year=2010|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-19452-5|chapter=2.1}}</ref>
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| ==Notes==
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| {{Reflist}}
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| {{DEFAULTSORT:Method Of Moments (Probability Theory)}}
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| [[Category:Probability theory]]
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Latest revision as of 23:00, 28 November 2014
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