Inverting Laplace transform

Question

From Williams' Probability w/ Martingales:

---

---

1. Are we allowed to switch derivative and integral as follows:

$$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = \int_{0}^{\infty} \frac{\partial}{\partial \lambda} e^{-\lambda x} f(x) $$

?

Why/Why not?

---

2. Assuming the $E[f(S_n)]$ equation is true, how does one prove the $f(y)$ equation?

That is, consider $E[f(S_n)]$ as a function of $\lambda$:

$$E[f(S_n)] = E[f(S_n)](\lambda) = \frac{(-1)^n (\lambda)^n L^{n-1}(\lambda)}{(n-1)!}$$

If $\lambda = \frac{n}{y}$, then

$$E[f(S_n)](\frac{n}{y}) = \frac{(-1)^n (\frac{n}{y})^n L^{n-1}(\frac{n}{y})}{(n-1)!}$$

How does one prove that

$$\lim_{n \to \infty} E[f(S_n)](\frac{n}{y}) \left(= \lim_{n \to \infty} \frac{(-1)^n (\frac{n}{y})^n L^{n-1}(\frac{n}{y})}{(n-1)!} \right) = f(y)?$$

Answer 1

For 1., the interchangeability follows from the Leibniz integral rule.

Part 2. is tricky, my attempt probably is filled with gaps, but here it is: let $\{X_i\}_{i=1}^n$ be iid exponential rvs of rate $n/y$, then $S_n\sim\mathrm{Erlang}(n,n/y)$, so $$\varphi_{S_n}(t)=\left(1-\frac{ity}{n}\right)^{-n}$$ But pointwise $$\lim_{n\to\infty}\left(1-\frac{ity}{n}\right)^{-n}=e^{ity},$$ which means the limiting distribution $S_n\to S$ degenerates at $y$, such that $S$ has Dirac delta as pdf: $$f_S(s)=\delta(s-y).$$ Hence, by portmanteau theorem, and the fact that $y>0$, $$\lim_{n\to\infty} E[f(S_n)]=E[f(S)]=\int_{0}^{\infty}f(s)\delta(s-y)\,ds=f(y).$$