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

\operatorname {E} (X^{k})\,

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

\lim _{n\to \infty }\operatorname {E} (X_{n}^{k})=\operatorname {E} (X^{k})\,

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

The method of moments was first introduced in 1891 by Pafnuty Chebyshev for proving the central limit theorem; although his proof is considered incomplete.^[2]^[3] Chebyshev cited earlier contributions by Irénée-Jules Bienaymé.^[3] More recently, the method has been applied by Eugene Wigner to prove Wigner's semicircle law, and has since found numerous applications in the theory of random matrices.^[4]

Notes

↑ Prokhorov, A.V. "Moments, method of (in probability theory)". In M. Hazewinkel (ed.). Encyclopaedia of Mathematics (online). ISBN 1-4020-0609-8. MR 1375697.
↑ Chebyshev, P. "Sur deux théorèmes relatifs aux probabilités" [On two theorems relating to probability]. Acta Mathematica (in French). 14: 305–315.
1 2 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. pp. 171–174. ISBN 978-0-387-87856-0. MR 2743162.
↑ 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.

[1] Prokhorov, A.V. "Moments, method of (in probability theory)". In M. Hazewinkel (ed.). Encyclopaedia of Mathematics (online). ISBN 1-4020-0609-8. MR 1375697.

[2] Chebyshev, P. "Sur deux théorèmes relatifs aux probabilités" [On two theorems relating to probability]. Acta Mathematica (in French). 14: 305–315.

[:0-3] 1 2 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. pp. 171–174. ISBN 978-0-387-87856-0. MR 2743162.

[4] 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.

[1]

[2]

[3]

[4]