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Given a compound distribution $$S:=\sum_{k=1}^{N} X_{i}$$ with

  1. $N$ is a discrete random variable with values in $\mathbb{N}$ with finite mean and variance.
  2. $X_{k}$ are non-negative iid random variables such that $\mathbb{E}\left[X_{i}\right]<\infty$ and $\sigma^{2}$:=Var($X_{i}$)<$\infty$.
  3. $N$ and $(X_{1},X_{2},\ldots)$ are independent.

Then $$\mathbb{E}\left[ S\right]= \mathbb{E}\left[ N\right]\mathbb{E}\left[ X_{1}\right]$$ as well as $$\text{Var}\left[ S\right]= \text{Var}\left[ N\right]\mathbb{E}\left[ X_{1}\right]^{2} + \mathbb{E}\left[ N\right]\text{Var}\left[ X_{1}\right].$$ Is there now a form of the central limit theorem applicable? In particular, are there conditions under which we can use normal approximation, i.e.

$$\frac{S-\mathbb{E}\left[S \right]}{\sqrt{\text{Var}\left[S \right]}}\approx \mathcal{N}(0,1)\; ?$$

$\mathbf{Update}$: Thanks a lot for the reference. Suppose that $\mathbb{E}\left[ X_{i}\right]=0$ and $N(t), t \geq 0$ is a family of positive, integer valued random variables, such that there is a $\theta >0$ $$ \frac{N(t)}{t}\stackrel{\mathbb{P}}{\rightarrow}\theta,\,\text{ as } t\to\infty. $$

According to Renyi's or Anscombe's Theorem, we then have \begin{align*} \frac{S_{N(t)}}{\sigma\sqrt{N(t)}} \xrightarrow[]{d} \mathcal{N}(0,1)\,\text{ as } t\to\infty \\ \frac{S_{N(t)}}{\sigma\sqrt{\theta\cdot t}} \xrightarrow[]{d} \mathcal{N}(0,1)\,\text{ as } t\to\infty, \\ \end{align*} which is different from the above normal approximation (using Wald's identity).

My question is now:

  1. Under which assumptions is the first/above normal approximation valid?

  2. What is the key difference between the two approximations respectively which one is preferrable? For example if $N(t)$ is a Poisson distribution?

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  • $\begingroup$ If $N\sim \text{Poisson}(\lambda)$ then you have a limit distribution as $\lambda\to \infty$. This is called the CLT for compound poisson processes. $\endgroup$
    – John
    Mar 2, 2018 at 14:55
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    $\begingroup$ Look at Anscombe's theorem (say, Th 2.3 here: uu.diva-portal.org/smash/get/diva2:438123/FULLTEXT01.pdf) $\endgroup$
    – NCh
    Mar 2, 2018 at 18:02

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