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 , , 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.
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
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]