absolute moments bounding (necessary and sufficient condition)
if and only if,βiβ₯0,iβπ
E[|X|k]β€E[|X|i]Mk-i βkβ₯i,kβπ |
Proof.
a) (E[|X|k]β€E[|X|i]Mk-i βΉ E[|X|k]β€Mk)
Itβs enough to take i=0 and the thesis follows easily.
b) (E[|X|k]β€Mk βΉE[|X|k]β€E[|X|i]Mk-i)
Let 1β€iβ€k (the case i=0 is trivial). Then, using Cauchy-Schwarz inequality N times, one has:
E[|X|k] | = | E[|X|i2|X|k-i2] | ||
β€ | E[|X|i]12E[|X|2k-i]12 | |||
= | E[|X|i]12E[|X|i2|X|2k-32i]12 | |||
β€ | E[|X|i](12+14)E[|X|4k-3i]14 | |||
β€ | E[|X|i](12+14+18)E[|X|(8k-7i)]18 | |||
β¦ | ||||
β€ | E[|X|i](βNm=112m)E[|X|2Nk-(2N-1)i]12N | |||
= | E[|X|i](1-12N)E[|X|2N(k-i)+i]12N | |||
β€ | E[|X|i](1-12N)M(k-i)+i2N, |
and since this must hold for any N, we obtain
E[|X|k]β€E[|X|i]Mk-i |
β
Title | absolute moments bounding (necessary and sufficient condition) |
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Canonical name | AbsoluteMomentsBoundingnecessaryAndSufficientCondition |
Date of creation | 2013-03-22 16:13:58 |
Last modified on | 2013-03-22 16:13:58 |
Owner | Andrea Ambrosio (7332) |
Last modified by | Andrea Ambrosio (7332) |
Numerical id | 5 |
Author | Andrea Ambrosio (7332) |
Entry type | Theorem |
Classification | msc 60E15 |