I have to prove the following.
Let $D_1,\ldots,D_n$ be a sequence of i.i.d. random variable and assume that $1\leq D_1 \leq m$, for some constant $m$. Let $f(t)$ be a smooth function on $[0,1]$ with bounded derivatives.
Prove that there exists a smooth function $g(t)$ on $[0,1]$, such that $$g(t)=\lim_{n\rightarrow \infty} \frac{1}{n} \log \mathbb{E}\left(\exp\left(\ell_nf\left(\frac{\ell_{[nt]}}{\ell_n}\right)\right)\right),$$ where $\ell_k=D_1+\ldots+D_k$ for all $1\leq k\leq n$.
It is also satisfied to provide a formula for the function $g$.
I observe that this problem is some how similar to the Large deviations theory. However, I have not found a good referrence for this problem.
Do you have any suggestion?