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 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

1. Prokhorov, A.V. "Moments, method of (in probability theory)". In M. Hazewinkel. Encyclopaedia of Mathematics (online). ISBN 1-4020-0609-8. MR 1375697.
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. ISBN 978-0-387-87856-0. MR 2743162.
3. 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.