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I'm confused about the behaviour to expect from a large sum of independent and identically distributed Poisson variables.

  • We know that a sum of $n$ Poisson variables of identical mean $\lambda$ is a Poisson variable of mean $n\lambda$.
  • We also know from the Central Limit Theorem that if $n$ is large, this sum should be approximately normal of mean $n\lambda$ and variance $n\lambda$ as well.

Now, what if we set $n\lambda = 1$ for instance?

Does it mean that the $\lambda$ mean parameter cannot anymore be considered as finite when $n$ is large, so we cannot apply the CLT anymore? Because obviously, a $\mathcal{P}oisson(1)$ is different from a $\mathcal{N}(1,1)$...

Context: the concrete situation is from a model of evolution of DNA sequences:

we take a sequence of $n$ sites, and at each site, the number of mutations in a given time follows a Poisson distribution, but we know that over the whole sequence (e.g 200-1000 sites), the mean will be like 5 mutations in 1 million years. I feel like I can't apply the CLT, although in a specific paper, the normality assumption is used to deduce confidence intervals.

PlasmaBinturong
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    A variety of approaches described at https://stats.stackexchange.com/questions/5347 for sums of Binomial distributions apply to this question, too. A search that includes [Berry Esseen](https://stats.stackexchange.com/search?q=Berry+esseen+poisson) is also useful: it turned up the duplicate post. – whuber Jun 28 '19 at 15:28
  • I didn't know about Berry-Esseen. But why is it applicable here? Do you consider that each site process has an expectation of 0 ? – PlasmaBinturong Jun 28 '19 at 15:57
  • The only Poisson variable with an expectation of zero is the constant number zero, which will contribute nothing to a sum. I wonder whether you might be confusing a Poisson *distribution* with a Poisson *process.* – whuber Jun 28 '19 at 17:02
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    Hmm, no I think I'm ok on that. I was just trying to understand why you linked my question with the Berry-Esseen theorem, but I just had an enlightenment now ^^ – PlasmaBinturong Jun 28 '19 at 17:14
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    If $n\to\infty$ while $\lambda\downarrow0$ so that $n\lambda$ remains fixed, then the hypotheses of the central limit theorem are not satisfied. You need to look at what those hypotheses are. $\qquad$ – Michael Hardy Jun 28 '19 at 17:17
  • @MichaelHardy That's what I was worried about yes. And also I needed something less fuzzy to define "finite", since this could be a valid approx when $n\lambda$ > 10 apparently. However in this case, I grasp that the approximation has nothing to do with the CLT, but is more a property of the Poisson distribution. – PlasmaBinturong Jul 01 '19 at 13:11
  • It does have to do with the CLT. $\qquad$ – Michael Hardy Jul 01 '19 at 23:11

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