We are going to show that is a normal distribution
for each fixed
, where
is a Brownian motion.
First, we are going to find out the distribution of . We can write
using the definition of Riemann integral,
If we write the increment on a segment by , then
Note that are iid random variables of
. Together with the fact
we have .
Next, we simply use change of variables to conclude the result as follows,
where .