probability theory Probability theory 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 expressing it through a set o ...

, the method of moments is a way of proving

convergence in distribution In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...

by proving convergence of a sequence of

moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...

sequences. Suppose ''X'' is a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

and that all of the moments :

\operatorname(X^k)\,

exist. Further suppose the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

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 In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ''μ'' to the sequences of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) ...

). 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. Chebyshe ...

for proving the

central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...

; 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 "E. P." Wigner ( hu, Wigner Jenő Pál, ; 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 con ...

to prove

Wigner's semicircle law The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...

, and has since found numerous applications in the theory of random matrices.

Notes

{{Reflist Moment (mathematics)