Marcinkiewicz–Zygmund inequality

Statement of the inequality

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

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

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