# How to decide if Central Limit Theorem is applicable on a sum of Poisson variables? [duplicate]

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.

• 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. – 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
• 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
• 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