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 $X_i$, $i=1,\dots ,n$, are independent random variables such that $E\left(X_i\right)=0$ and $E\left({\left|X_i\right|}^p\right)<+\mathrm{\infty }$, $1\le p<+\mathrm{\infty }$, then

$A_{p}E\left({\left(\sum _{i=1}^n{\left|X_i\right|}^2\right)}_{}^{p/2}\right)\le E\left({\left|\sum _{i=1}^n{X}_i\right|}^p\right)\le B_{p}E\left({\left(\sum _{i=1}^n{\left|X_i\right|}^2\right)}_{}^{p/2}\right)$

where $A_p$ and $B_p$ are positive constants, which depend only on $p$ and not on the underlying distribution of the random variables involved.

The second-order case

In the case $p=2$, the inequality holds with $A_2=B_2=1$, and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If $E\left(X_i\right)=0$ and $E\left({\left|X_i\right|}^2\right)<+\mathrm{\infty }$, then

$\mathrm{V}\mathrm{a}\mathrm{r}\left(\sum _{i=1}^n{X}_i\right)=E\left({\left|\sum _{i=1}^n{X}_i\right|}^2\right)=\sum _{i=1}^n\sum _{j=1}^{n}E\left(X_i{\bar{X}}_j\right)=\sum _{i=1}^{n}E\left({\left|X_i\right|}^2\right)=\sum _{i=1}^n\mathrm{V}\mathrm{a}\mathrm{r}\left(X_i\right).$

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.