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.

Theorem [1][2] If

{\displaystyle \textstyle X_{i}}

, are independent random variables such that

are positive constants, which depend only on

and not on the underlying distribution of the random variables involved.

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

, then 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.