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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 X i {\displaystyle \textstyle X_{i}} , i = 1 , … , n {\displaystyle \textstyle i=1,\ldots ,n} , are independent random variables such that E ( X i ) = 0 {\displaystyle \textstyle E\left(X_{i}\right)=0} and E ( | X i | p ) < + ∞ {\displaystyle \textstyle E\left(\left\vert X_{i}\right\vert ^{p}\right)<+\infty } , 1 ≤ p < + ∞ {\displaystyle \textstyle 1\leq p<+\infty } , then
A p E ( ( ∑ i = 1 n | X i | 2 ) p / 2 ) ≤ E ( | ∑ i = 1 n X i | p ) ≤ B p E ( ( ∑ i = 1 n | X i | 2 ) p / 2 ) {\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 A p {\displaystyle \textstyle A_{p}} and B p {\displaystyle \textstyle B_{p}} are positive constants, which depend only on p {\displaystyle \textstyle p} and not on the underlying distribution of the random variables involved.
The second-order case In the case p = 2 {\displaystyle \textstyle p=2} , the inequality holds with A 2 = B 2 = 1 {\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 E ( X i ) = 0 {\displaystyle \textstyle E\left(X_{i}\right)=0} and E ( | X i | 2 ) < + ∞ {\displaystyle \textstyle E\left(\left\vert X_{i}\right\vert ^{2}\right)<+\infty } , then
V a r ( ∑ i = 1 n X i ) = E ( | ∑ i = 1 n X i | 2 ) = ∑ i = 1 n ∑ j = 1 n E ( X i X ¯ j ) = ∑ i = 1 n E ( | X i | 2 ) = ∑ i = 1 n V a r ( X i ) . {\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]
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