Marcinkiewicz–Zygmund inequality explained

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

style X _{i} , style i =1 ,\ldots,n , are independent random variables such that style E\left( X _{i} \right) =0 and style E\left( \left\vert X _{i} \right\vert ^{p} \right) < +inf ty , style 1 \leq p< +inf ty , thenA _{p} E\left( \left( \sum ^{n} _{i =1} \left\vert X _{i} \right\vert ^{2} \right) _{{} }^\right) \leq E\left(\left\vert \sum_^X_\right\vert ^\right) \leq B_E\left(\left(\sum_^\left\vert X_\right\vert ^\right) _^\right)

where

style A _{p} and style B _{p} are positive constants, which depend only on style p and not on the underlying distribution of the random variables involved.

The second-order case

In the case

style p =2 , the inequality holds with style 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 style E\left( X _{i} \right) =0 and style E\left( \left\vert X _{i} \right\vert ^{2} \right) < +inf ty , thenVar \left(\sum ^{n} _{i =1} X _{i} \right) = E\left( \left\vert \sum ^{n} _{i =1} X _{i} \right\vert ^{2} \right) = \sum ^{n} _{i =1} \sum ^{n} _{j =1} E\left( X _{i} \overline{X} _{j} \right) = \sum ^{n} _{i =1} E\left( \left\vert X _{i} \right\vert ^{2} \right) = \sum ^{n} _{i =1} 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]

Notes

1. J. Marcinkiewicz and A. Zygmund. Sur les fonctions indépendantes. Fund. Math., 28:60 - 90, 1937. Reprinted in Józef Marcinkiewicz, Collected papers, edited by Antoni Zygmund, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964, pp. 233 - 259.
2. Yuan Shih Chow and Henry Teicher. Probability theory. Independence, interchangeability, martingales. Springer-Verlag, New York, second edition, 1988.
3. R. Ibragimov and Sh. Sharakhmetov. Analogues of Khintchine, Marcinkiewicz - Zygmund and Rosenthal inequalities for symmetric statistics. Scandinavian Journal of Statistics, 26(4):621 - 633, 1999.