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. It is a special case of the Burkholder-Davis-Gundy inequality in the case of discrete-time martingales.

Statement of the inequality

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

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

where ${\displaystyle \textstyle A_{p}}$ and ${\displaystyle \textstyle B_{p}}$ are positive constants, which depend only on ${\displaystyle \textstyle p}$ and not on the underlying distribution of the random variables involved.

The second-order case

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

${\displaystyle \mathrm {Var} \left(\sum _{i=1}^{n}X_{i}\right)=E\left(\left\vert \sum _{i=1}^{n}X_{i}\right\vert ^{2}\right)=\sum _{i=1}^{n}\sum _{j=1}^{n}E\left(X_{i}{\overline {X}}_{j}\right)=\sum _{i=1}^{n}E\left(\left\vert X_{i}\right\vert ^{2}\right)=\sum _{i=1}^{n}\mathrm {Var} \left(X_{i}\right).}$

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 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.
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.