# Method of moments (probability theory)

In probability theory, the **method of moments** is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences.^{[1]} Suppose *X* is a random variable and that all of the moments

- [math]\displaystyle{ \operatorname{E}(X^k)\, }[/math]

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
(cf. the problem of moments). If

- [math]\displaystyle{ \lim_{n\to\infty}\operatorname{E}(X_n^k) = \operatorname{E}(X^k)\, }[/math]

for all values of *k*, then the sequence {*X*_{n}} converges to *X* in distribution.

The method of moments was introduced by Pafnuty Chebyshev for proving the central limit theorem; Chebyshev cited earlier contributions by Irénée-Jules Bienaymé.^{[2]} More recently, it has been applied by Eugene Wigner to prove Wigner's semicircle law, and has since found numerous applications in the theory of random matrices.^{[3]}

## Notes

- ↑ Prokhorov, A.V.. "Moments, method of (in probability theory)". in M. Hazewinkel.
*Encyclopaedia of Mathematics (online)*. ISBN 1-4020-0609-8. http://encyclopediaofmath.org/index.php?title=Moments,_method_of_(in_probability_theory)&oldid=47882. - ↑ Fischer, H. (2011). "4. Chebyshev's and Markov's Contributions.".
*A history of the central limit theorem. From classical to modern probability theory.*. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. ISBN 978-0-387-87856-0. - ↑ Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). "2.1".
*An introduction to random matrices.*. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.

Original source: https://en.wikipedia.org/wiki/Method of moments (probability theory).
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