TITLE: Poisson random variable is not sub gaussian
QUESTION [6 upvotes]: 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!

REPLY [5 votes]: 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.