relation between almost surely absolutely bounded random variables and their absolute moments
1) i.e. is absolutely bounded almost surely;
Then by hypothesis
Then we have obviously (in fact, if ) and (in fact, let ; let ; then , that is ); this means that
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 ,
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|
|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)|
|Author||Andrea Ambrosio (7332)|