From Williams' Probability w/ Martingales:

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  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?

  1. 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

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

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    $\begingroup$ Cross-posted at math.stackexchange.com/q/1515987/59351. I've lost count of the number of times I've asked you not to re-post your old questions from Mathematics here - would you please stop doing it? It's against SE policy (NB "Just to be 100% clear, copy-pasting a question across sites with no changes is considered abusive behavor" from J.A. himself) & explicitly discouraged on our help pages. If you've got a question on Mathematics you think would benefit from ... $\endgroup$ – Scortchi - Reinstate Monica Jul 10 '16 at 13:38
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    $\begingroup$ ... answers with a statistical rather than a mathematical perspective, then when you ask it here you need to adapt it to make that clear & link to the original question. Not linking in particular is discourteous to people who might wish to take into account answers to, comments on, or just the existence of, the original question when deciding if or how to answer its duplicate. $\endgroup$ – Scortchi - Reinstate Monica Jul 11 '16 at 10:32
  • $\begingroup$ @Scortchi Very well. Deleted. $\endgroup$ – BCLC Jul 12 '16 at 0:09
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    $\begingroup$ What's deleted? This question & the duplicate on Mathematics can't be deleted now because they've already got answers on both sites. For your other duplicated questions please decide which site you want them on; or if they've already got answers on both, link to the duplicate (when I or someone else haven't yet added a link in a comment). $\endgroup$ – Scortchi - Reinstate Monica Jul 12 '16 at 8:50

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 $y>0$, $$\lim_{n\to\infty} E[f(S_n)]=E[f(S)]=\int_{0}^{\infty}f(s)\delta(s-y)\,ds=f(y).$$


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