TITLE: Tails of the normal distribution
QUESTION [6 upvotes]: I'm currently reading Roman Vershynin's High-Dimensional Probability. For Proposition $2.1.2$, I wonder how the lower bound is obtained. I understand that the lower bound is correct, but I don't know where the term $-3x^{-4}$ comes from. Thanks.

REPLY [4 votes]: It just removes a term to help the proof. Namely, for any $x>0$,
$$
\frac{1}{\sqrt{2\pi}} e^{-x^2/2} \geq (1-3x^{-4})\cdot \frac{1}{\sqrt{2\pi}} e^{-x^2/2}
$$
since you just remove something positive from the LHS; and then you can use the "rabbit-out-of-the-hat" identity he gives to give an exact expression for the integral of the RHS.