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


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