# Gaussian concentration/isoperimetric inequality with correlated Gaussian measure

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 , 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 $$F$$.

It seems that a more general version of Gaussian isoperimetric inequality (i.e. Bakry Emery) applies to distribution with density function of the form: $$\propto e^{-H(t)}dx$$ where $$\nabla^2H\ge K,K\in\mathbb{R}^+$$ (this means $$\ge KI_n$$). Bakry-Emery then states: the log-Sobolev inequality holds with $$\gamma \leq \frac{2}{K}$$ : $$S(f) \leq \frac{2}{K} D(\sqrt{f})$$ where $$S$$ is entropy, $$D$$ is Dirichlet form. Concentration result like the one in my problem statement will follow from Herbst argument with Lipschitz test functions.
In the particular log-sum-expo case I gave, i.e. for correlated Gaussian, $$\nabla^2H=\Sigma^{-1}>\lambda_n(\Sigma^{-1})=(\lambda_1(\Sigma))^{-1}\ge n/C^2$$ by Schur-Horn inequality. So concentration holds with subgaussian norm O($$C$$)