relation between almost surely absolutely bounded random variables and their absolute moments
Let {Ω,E,P} a probability space and let X be a random
variable
; then, the following are equivalent
:
1) Pr{|X|≤M}=1 i.e. X is absolutely bounded almost surely;
2) E[|X|k]≤Mk ∀k≥1,k∈N
Proof.
1) ⟹ 2)
Let’s define
F={ω∈Ω:|X(ω)|>M}; |
Then by hypothesis
Pr{Ω\F}=1 |
and
Pr{F}=0. |
We have:
E[|X|k] | = | ∫Ω|X|k𝑑P | ||
= | ∫Ω\F|X|k𝑑P+∫F|X|k𝑑P | |||
= | ∫Ω\F|X|k𝑑P | |||
≤ | ∫Ω\FMk𝑑P | |||
= | MkPr{Ω\F}=Mk. |
2) ⟹ 1)
Let’s define
F | = | {ω∈Ω:|X(ω)|>M} | ||
Fn | = | {ω∈Ω:|X(ω)|>M+1n} ∀n≥1. |
Then we have obviously Fn⊆Fn+1 (in fact, if ω∈Fn⟹|X(ω)|>M+1n>M+1n+1⟹ω∈Fn+1) and F=⋃∞n=1Fn (in fact, let ω∈F; let N=⌈1|X(ω)|-M⌉; then |X(ω)|>M+1N, that is ω∈FN); this means that
F=lim |
in the meaning of sets sequences convergence (http://planetmath.org/SequenceOfSetsConvergence).
So the continuity from below property (http://planetmath.org/PropertiesForMeasure) of probability can be applied:
Now, for any ,
that is
so that the only acceptable value for is
whence the thesis. ∎
Acknowledgements: due to helpful discussions with Mathprof.
Title | relation between almost surely absolutely bounded random variables and their absolute moments |
---|---|
Canonical name | RelationBetweenAlmostSurelyAbsolutelyBoundedRandomVariablesAndTheirAbsoluteMoments |
Date of creation | 2013-03-22 16:14:33 |
Last modified on | 2013-03-22 16:14:33 |
Owner | Andrea Ambrosio (7332) |
Last modified by | Andrea Ambrosio (7332) |
Numerical id | 8 |
Author | Andrea Ambrosio (7332) |
Entry type | Theorem |
Classification | msc 60A10 |