Let $X=(X_1,\ldots,X_n)$ be a random vector in $\mathbb R^n$ such that
- Gaussian marginals, i.e $X_i$ has distribution $N(0,1)$ for every $i \in \{1,\ldots,n\}$
- Exchangeable coordinates, i.e for every $I \subseteq \{1,\ldots,n\}$, the distribution of the random sub-vector $(X_i)_{i \in I}$ is permutation-invariant.
Let, for $k \in 1,2,\ldots$, let $M^{(k)} := \|X\|_2^k$, where $\|X\|_2 := (\sum_{i=1}^n |X_i|^2)^{1/2}$.
Question. Are the any concentration and anti-concentration inequalities for the $M^{(k)}$'s ?
I'm particularly interested in the cases $k = 1,2$.
Note. By (anti-)concentration inequality, I mean upper (resp. lower) bounds on $\mathbb P(|M^{(k)} - \mathbb E[M^{(k)}]| > \epsilon)$, for every $\epsilon \ge 0$.