are finite, I have still some problems with proving that
My first problem is how toexploit the knowledge about the tail distribution of supremum to calculate the
-th moment. I know the formula
is the cdf of X. But is there any similar formula valid for higher moments?
The second problem is that the bound in Borell's theorem holds for
and we don't know what happens if
I have been trying to solve the problem without Borell's theorem but I gave it up.
Answer to your first question:
follows from a simple change of variable in Cavalieri's formula (or Tonelli's theorem).
Answer to your second question: Since the process has mean 0 (and assumed to have continuous paths?), you don't really have to worry about u<0 --- remember
and reflect the process will give you an immediate bound.