In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Marcinkiewicz–Zygmund inequality, named after
Józef Marcinkiewicz and
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...
, gives relations between
moments of a collection of
independent random variables
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
. It is a generalization of the rule for the sum of
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
s of independent random variables to moments of arbitrary order. It is a special case of the
Burkholder-Davis-Gundy inequality In mathematics, quadratic variation is used in the analysis of stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic pr ...
in the case of discrete-time martingales.
Statement of the inequality
Theorem
[J. Marcinkiewicz and A. Zygmund. Sur les fonctions indépendantes. ''Fund. Math.'', 28:60–90, 1937. Reprinted in Józef Marcinkiewicz, ''Collected papers'', edited by Antoni Zygmund, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964, pp. 233–259.
][Yuan Shih Chow and Henry Teicher. ''Probability theory. Independence, interchangeability, martingales''. Springer-Verlag, New York, second edition, 1988.
] If
,
, are independent random variables such that
and
,
, then
:
where
and
are positive constants, which depend only on
and not on the underlying distribution of the random variables involved.
The second-order case
In the case
, the inequality holds with
, and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If
and
, then
:
See also
Several similar moment inequalities are known as
Khintchine inequality In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N complex ...
and
Rosenthal inequalities, and there are also extensions to more general symmetric
statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
s of independent random variables.
[R. Ibragimov and Sh. Sharakhmetov. Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal inequalities for symmetric statistics. ''Scandinavian Journal of Statistics'', 26(4):621–633, 1999.]
Notes
{{DEFAULTSORT:Marcinkiewicz-Zygmund inequality
Statistical inequalities
Probabilistic inequalities
Probability theorems
Theorems in functional analysis