Famous Gaussian concentration inequality states that:

If $\mathrm{F}$ is 1 -Lip, and $\mathbb{E} F(X)=0,$ and $X=(X_1,...,X_n) \sim N\left(0, I_{n}\right),$ then we have for some absolute constant $C>0$ $$ \|F(X)\|_{\psi_{2}} \leq C $$ uniformly for all $\mathrm{n}$.

I always thought the independent condition for $X_1,...,X_n$ here is indispensable. But today I came across a special case: for finite measure $G(\cdot)$ on $\mathbb{N}$, and $F(X):=\log \sum_i \exp(X_i)G(i)$. If we have $\sup_i \mathbb{E}X_i^2 <C^2$, then

$$ \|F(X)-\mathbb{E}F(X)\|_{\psi_{2}} \leq C $$ even when $X_1,X_2,...$ are correlated. I wonder if there is a generalization of such concentration inequality for correlated Gaussians for a class of function $F$.