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 $F$.


1 Answer 1


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($C$)


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