I have a sum of the form:
$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$
where $f(x)$ and $g(x)$ are smooth functions defined for $0\le x \le 1$. I am interested in the Asymptotic behaviour of $S_n$ as $n\rightarrow +\infty$.
What I have tried is to replace the sum by an integral:
$$S_n \approx\int_0^1 \mathrm{e}^{n f(x)} g(x) \mathrm{d}x$$
and then I was thinking of doing a saddle-point approximation (Laplace's method), but I am not sure of the validity of replacing the sum by the integral. It looks like a Riemann sum, except for the $n$ multiplying $f$ in the exponent.
Any suggestions appreciated.
If it makes your life easier assume that $f(x)$ has a single maximum inside the interval $x\in(0,1)$.
I'd be happy if someone could at least point out some reference to the literature where sums like this have popped out before. I have the feeling that this has been treated before.
Edit, TLTR. The summarized version of the question is: Prove or disprove that $S_n / I_n \rightarrow c$ for some constant $c$ as $n\rightarrow \infty$, where
$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n), \quad I_n = \int_0^1 \mathrm{e}^{n f(x)} g(x) \mathrm{d}x$$
Moreover, is the difference between $I_n$ and $S_n$ exponentially decreasing in $n$? That is, $|I_nc - S_n| / S_n = \mathcal{O}(n^p e^{-qn})$ for some numbers $p,q$?