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 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 $\textstyle p$ and not on the underlying distribution of the random variables involved.

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.