proof of Bernstein inequalities
1) By Chernoff-Cramèr bound (http://planetmath.org/ChernoffCramerBound), we have:
and, keeping in mind hypotheses a) and b),
Now, if , we obtain
which, once plugged into the bounds, yields
Observing that , one gets:
Plugging in the bound expression, the sub-optimal yet more easily manageable formula is obtained:
(see here (http://planetmath.org/ASimpleMethodForComparingRealFunctions) for an easy way, which can be used with )
2) To prove this more specialized statement let’s recall that the condition
implies that, for all ,
(See here (http://planetmath.org/RelationBetweenAlmostSurelyAbsolutelyBoundedRandomVariablesAndTheirAbsoluteMoments) for a proof.)
Now, it’s enough to verify that the condition
imply both conditions a) and b) in part 1).
Indeed, part a) is obvious, while for part b) one happens to have:
(see here (http://planetmath.org/AbsoluteMomentsBoundingNecessaryAndSufficientCondition) for a proof).
Let’s find a value for such that , thus satisfying part b) of the hypotheses.
After simplifying, we have to study the inequality
for any . Let’s proceed by induction. For , we have
which confirms the validity of the choice , which has to be plugged into the former bound to obtain the new one.
[to be continued…]
|Title||proof of Bernstein inequalities|
|Date of creation||2013-03-22 16:09:32|
|Last modified on||2013-03-22 16:09:32|
|Owner||Andrea Ambrosio (7332)|
|Last modified by||Andrea Ambrosio (7332)|
|Author||Andrea Ambrosio (7332)|