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I am reading a chapter on concentration inequalities, and I am struggling to make connections on sub-Gaussian random variables. A random variable is sub-Gaussian if there exists $C > 0$ such that $P(|X| \geq t) \leq 2\exp(-t^2/C^2)$. The text gives numerous equivalent conditions to this one.

I have an exercise: If $X \sim Poi(\lambda)$, show that $X$ is not sub-Gaussian.

My setup is $P(|X| \geq t) = P(X \geq t) = \sum_{k=t}^{\infty} \frac{e^{-\lambda}\lambda^{k}}{k!}$ since the Poisson random variable is supported on nonnegative integers. My first thought was to use Stirling's formula on the $k!$, but I wouldn't know how to sum the resulting series. I appreciate any help on this question!

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Hint

I don't think you need to do any summing. You have $$ P(X\ge t) > \frac{e^{-\lambda}\lambda^t}{t!}.$$ You can apply Stirling to this to get an asymptotic lower bound and show it decays slower than any Gaussian.

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