I have a queston about the convergence in law of the following stochastic processe:
$\left\{I_t=\left(\int_0^te^{B_s}ds\right)^{1/\sqrt{t}}\right\}_{t\geq 0}$
with $\{B_t\}_{t\geq 0}$ is a standrad brownian motion.
Prove that $I_t\rightarrow e^{|N|}$ in law, where $N$ has the gaussian distribution $N(0,1)$.
I have tried by scaling property of brownian motion, but it does not work. I try also with the Laplace tranformation, but it is really difficult to discrible $I_t$'s transformation. Does someone have an idea? Thanks a lot!