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The CDF of a Poisson distribution with rate parameter $\lambda$ is $$ P(n;\lambda)=\sum_{k=0}^n \frac{\lambda^ke^{-\lambda}}{k!}. $$ As $n$ goes to infinity, the CDF would certainly approach 1.

Now, consider the case when the rate parameter is $\xi n$ with $\xi\in(0,1)$ being a given constant. According to the last answer in https://math.stackexchange.com/questions/160248/evaluating-lim-limits-n-to-infty-e-n-sum-limits-k-0n-fracnkk, the following limit should be 1: $$ \lim_{n\to\infty}P(n;\xi n)=\lim_{n\to\infty} \sum_{k=0}^n\frac{(n\xi)^ke^{-n\xi}}{k!}=1. $$ Is it possible to check the convergence rate of the above limit? More specifically, I wonder whether the convergence rate is faster than $n$, i.e., whether $n[1-P(n;\xi n)]=O(1)$?

I checked using software that the convergence rate is faster than $n$, but I don't know how to show it rigorously. Can anyone provide some hints and insights? Thanks!

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