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,, 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