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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!

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@ Higgs88 : You should think about Laplace Method (the one that talk about $L^P$ convergence when $p\to\infty$ of a suitable random variable (or measurable function), by the way scaling property is clearly a step in the process. Best regards –  TheBridge Oct 13 '12 at 21:30

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