proof of Chernoff-Cramer bound
Since this expression is valid for any , the best bound is obtained taking the supremum:
which proves part c).
To prove part a), let’s observe that and that
that is, is the infimum point for and consequently for , so as a conclusion
b) Let be fixed and let be the supremum point for ; we have to show that .
By differentiation, .
The convexity of follows from the fact that is the supremum of the linear (and hence convex) functions and so must be convex itself.
Eventually, in to prove that is an increasing function, let’s note that
and that, by Taylor formula with Lagrange form remainder, for a
since by convexity and by hypotheses.
|Title||proof of Chernoff-Cramer bound|
|Date of creation||2013-03-22 16:09:05|
|Last modified on||2013-03-22 16:09:05|
|Owner||Andrea Ambrosio (7332)|
|Last modified by||Andrea Ambrosio (7332)|
|Author||Andrea Ambrosio (7332)|