In the lecture notes https://galton.uchicago.edu/~lalley/Courses/386/Concentration.pdf , the following corollary of McDiarmid's inequality is proven (Corollary 2.1): if $X_1,\dots,X_N$ are standard normal random variables, and $g:\mathbb{R}^N \to \mathbb{R}$ is Lipschitz in each variable separately (i.e. $|g(X_1,\dots,X_N) - g(X_1,\dots,X_{i-1},X'_i,X_{i+1},\dots,X_N)| \le |X_i - X'_i|$ for all $i$), then
$\Pr[|g(X_1,\dots,X_N) - \mathbb{E}[g(X_1,\dots,X_N)]| \ge t] \le 2e^{-2t^2/N}$.
Although this inequality seems like it should be fairly standard, I was not able to find it proven in a textbook or paper. My questions are:
- What is a good reference to cite for this?
- Is the constant of 2 in the exponent correct? I'm confused because it seems like taking $N=1$, $g(X)=X$, we would obtain stronger tail bounds than I would expect for a normal random variable (I would have expected $\Pr[|g(X)| \ge t] \le 2e^{-t^2/2}$).