Each
is clearly
-measureable and finite a.s. (Hence
). Therefore we only need to verify the martingale property. That is, we want to show
We can assert that
exists and is finite since each
almost surely. Therefore, in order to make
a martingale, we must have
.
First let us find the distribution of
:
Thus by the chain rule, our random variable
has probability density function
So then
Now integrate the remaining integral by parts letting
. We get:
Repeat integration by parts another
times and we get
Let be independent random variables such that
(a) Find the characteristic function of .
(b) Show that converges in distribution to a non-degenerate random variable.
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Then by independence, we have