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 , , are independent random variables such that and , , then

where and are positive constants, which depend only on .

The second-order case[edit]

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[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.