“Marcinkiewicz–Zygmund inequality”的意思、由来-开放百科全书

词条

Marcinkiewicz–Zygmund inequality

释义

Statement of the inequality
The second-order case
See also
Notes

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 , , 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

Notes

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.

4 : Statistical inequalities|Probabilistic inequalities|Probability theorems|Theorems in functional analysis

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Statement of the inequality

The second-order case

See also

Notes