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Marcinkiewicz–Zygmund inequality

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In mathematics, the MarcinkiewiczZygmund 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.

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Statement of the inequality

Theorem [1][2] If , , are independent random variables such that and , , then

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

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The second-order case

In the case , the inequality holds with , and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If and , then

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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]

Notes

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