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.

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    $\begingroup$ A variety of approaches described at stats.stackexchange.com/questions/5347 for sums of Binomial distributions apply to this question, too. A search that includes Berry Esseen is also useful: it turned up the duplicate post. $\endgroup$ – whuber Jun 28 '19 at 15:28
  • $\begingroup$ 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 ? $\endgroup$ – PlasmaBinturong Jun 28 '19 at 15:57
  • $\begingroup$ 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. $\endgroup$ – whuber Jun 28 '19 at 17:02
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    $\begingroup$ 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 ^^ $\endgroup$ – PlasmaBinturong Jun 28 '19 at 17:14
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    $\begingroup$ 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. $\qquad$ $\endgroup$ – Michael Hardy Jun 28 '19 at 17:17