probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

, the method of moments is a way of proving

convergence in distribution In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...

by proving convergence of a sequence of moment sequences. Suppose ''X'' is a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

and that all of the moments :

\operatorname(X^k)\,

exist. Further suppose the

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

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_\operatorname(X_n^k) = \operatorname(X^k)\,

for all values of ''k'', then the sequence converges to ''X'' in distribution. The method of moments was introduced by

Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...

for proving the

central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...

; Chebyshev cited earlier contributions by

Irénée-Jules Bienaymé Irénée-Jules Bienaymé (; 28 August 1796 – 19 October 1878) was a French statistician. He built on the legacy of Laplace generalizing his least squares method. He contributed to the fields of probability and statistics, and to their applicat ...

. More recently, it has been applied by

Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...

to prove Wigner's semicircle law, and has since found numerous applications in the theory of random matrices.

Notes

{{Reflist Moments (mathematics)