Marcinkiewicz–Zygmund inequality

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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.

Statement of the inequality[edit]

Theorem [1][2] If \textstyle x_{i}, \textstyle i=1,\ldots,n, are independent random variables such that \textstyle E\left( x_{i}\right)  =0 and \textstyle E\left(  \left\vert x_{i}\right\vert ^{p}\right) <+\infty, \textstyle 1\leq p<+\infty,

 A_{p}E\left(  \left(  \sum_{i=1}^{n}\left\vert x_{i}\right\vert ^{2}\right) _{{}}^{p/2}\right)  \leq E\left(  \left\vert \sum_{i=1}^{n}x_{i}\right\vert ^{p}\right)  \leq B_{p}E\left(  \left(  \sum_{i=1}^{n}\left\vert x_{i}\right\vert ^{2}\right)  _{{}}^{p/2}\right)

where \textstyle A_{p} and \textstyle B_{p} are positive constants, which depend only on \textstyle p.

The second-order case[edit]

In the case \textstyle p=2, the inequality holds with \textstyle 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 \textstyle E\left( x_{i}\right)  =0 and \textstyle E\left(  \left\vert x_{i}\right\vert ^{2}\right) <+\infty, then

 \mathrm{Var}\left(\sum_{i=1}^{n}x_{i}\right)=E\left(  \left\vert \sum_{i=1}^{n}x_{i}\right\vert ^{2}\right)  =\sum_{i=1}^{n}\sum_{j=1}^{n}E\left( x_{i}\overline{x}_{j}\right)  =\sum_{i=1}^{n}E\left(  \left\vert x_{i}\right\vert ^{2}\right)  =\sum_{i=1}^{n}\mathrm{Var}\left(x_{i}\right).

See also[edit]

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[edit]

  1. ^ J. Marcinkiewicz and A. Zygmund. Sur les foncions independantes. 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.