Marcinkiewicz–Zygmund inequality

In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order.

Statement of the inequality

Theorem ^[1][2] If $\textstyle x_{i}$, $\textstyle i=1,\ldots,n$, are independent random variables such that $\textstyle E\left( x_{i}\right) =0$ and $\textstyle E\left( \left\vert x_{i}\right\vert ^{p}\right) <+\infty$, $\textstyle 1\leq p<+\infty$, then

$A_{p}E\left( \left( \sum_{i=1}^{n}\left\vert x_{i}\right\vert ^{2}\right) _{{}}^{p/2}\right) \leq E\left( \left\vert \sum_{i=1}^{n}x_{i}\right\vert ^{p}\right) \leq B_{p}E\left( \left( \sum_{i=1}^{n}\left\vert x_{i}\right\vert ^{2}\right) _{{}}^{p/2}\right)$

where $\textstyle A_{p}$ and $\textstyle B_{p}$ are positive constants, which depend only on $\textstyle p$.

The second-order case

In the case $\textstyle p=2$, the inequality holds with $\textstyle A_{2}=B_{2}=1$, and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If $\textstyle E\left( x_{i}\right) =0$ and $\textstyle E\left( \left\vert x_{i}\right\vert ^{2}\right) <+\infty$, then

$\mathrm{Var}\left(\sum_{i=1}^{n}x_{i}\right)=E\left( \left\vert \sum_{i=1}^{n}x_{i}\right\vert ^{2}\right) =\sum_{i=1}^{n}\sum_{j=1}^{n}E\left( x_{i}\overline{x}_{j}\right) =\sum_{i=1}^{n}E\left( \left\vert x_{i}\right\vert ^{2}\right) =\sum_{i=1}^{n}\mathrm{Var}\left(x_{i}\right).$

See also

Several similar moment inequalities are known as Khintchine inequality and Rosenthal inequalities, and there are also extensions to more general symmetric statistics of independent random variables.^[3]

Notes

1. J. Marcinkiewicz and A. Zygmund. Sur les foncions independantes. Fund. Math., 28:60–90, 1937. Reprinted in Józef Marcinkiewicz, Collected papers, edited by Antoni Zygmund, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964, pp. 233–259.
2. Yuan Shih Chow and Henry Teicher. Probability theory. Independence, interchangeability, martingales. Springer-Verlag, New York, second edition, 1988.
3. 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.