proof of Doob’s inequalities
and if is a martingale or nonnegative submartingale then,
for every and .
First, let us consider the case where is finite. The first time at which ,
However, if and only if giving,
where the supremum is understood to be over . Now suppose that is countable. Then choose finite subsets which increase to as goes to infinity. Replacing by in inequality (4) and using the monotone convergence theorem to take the limit extends (4) to arbitrary uncountable index sets. Then, inequality (1) follows immediately from (4).
Now, suppose that is a martingale. Jensen’s inequality gives
For , multiply (4) by and integrate up to some limit ,
The left hand side of this inequality can be computed by commuting the order of integration with respect to and (Fubini’s theorem),
The right hand side of (5) can be computed similarly,
Putting these back into (5),
Now let , so that are conjugate (http://planetmath.org/ConjugateIndex) and the Hölder inequality gives
Substituting into (6), the finite term cancels to get
and the result follows by letting increase to infinity.
|Title||proof of Doob’s inequalities|
|Date of creation||2013-03-22 18:39:55|
|Last modified on||2013-03-22 18:39:55|
|Last modified by||gel (22282)|