Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.

Now, I was wondering whether we could expect an inequality like the one we expect in the case where $X$ had iid $\mathcal{N}(0,1)$ components, i.e. in this other extreme case, can we expect that for any Lipschitz function $f: \mathbb{R}^n \to \mathbb{R},$ we have:

$$ P\left[ |f(X) - Ef(X)| \ge t \right] \le 2 exp \left( - \frac{ct^2}{||f||_{Lip}^2} \right), $$

where $c > 0$ is an absolute constant.

P.S. I might add a similar question on what happens to the above probability when $n \to \infty,$ but I'm still thinking about the question.

P.P.S. if you could cite some reference, that'd be very useful too. I do have access to Michel Lédoux's great book, but I find it a bit difficult for myself to quickly navigate through and get the right inequality I want (but that's just me!).

Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.

Now, I was wondering whether we could expect an inequality like the one we expect in the case where $X$ had iid $\mathcal{N}(0,1)$ components, i.e. in this other extreme case, can we expect that for any Lipschitz function $f: \mathbb{R}^n \to \mathbb{R},$ we have for each $t \ge 0:$

$$ P\left[ |f(X) - Ef(X)| \ge t \right] \le 2 exp \left( - \frac{ct^2}{||f||_{Lip}^2} \right), $$

where $c > 0$ is an absolute constant.

P.S. I might add a similar question on what happens to the above probability when $n \to \infty, t $ is fixed, but I'm still thinking about the question.

P.P.S. if you could cite some reference, that'd be very useful too. I do have access to Michel Lédoux's great book, but I find it a bit difficult for myself to quickly navigate through and get the right inequality I want (but that's just me!).