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. Suppose X is a random variable and that all of the moments

$\mathrm{E}(X^k)$

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

$\underset{n\to \mathrm{\infty }}{\lim }\mathrm{E}(X_n^k)=\mathrm{E}(X^k)$

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. 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.