Method of moments (probability theory)

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See method of moments (statistics) for an account of a method of parameter estimation.

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

\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 {Xn} converges to X in distribution.

The method of moments is especially useful for proving limits theorems for random matrices with independent entries, such as Wigner's semi-circle law.