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Let $X_n$ be a set of $N$ subgaussian random variables, not necessarily independent, with $E\exp(\lambda X_n) \le \exp(\lambda^2/2)$. Let $X=(X_1,\ldots, X_N)$ and $f:\mathbb R^N \rightarrow \mathbb R$ be a 1-Lipschitz function. Is it possible to establish a concentration inequality of the type:

$$P(|f(X)-E(f(X))| > \epsilon) \le \exp(-c\epsilon^2)$$

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  • $\begingroup$ $f$ is 1-Lipschitz under what metric? I don't think you actually need $X_n$ to be subgaussian. On the other hand, if they aren't independent, you'll definitely need some control over how strongly dependent they are (see mixing conditions). $\endgroup$
    – Aryeh Kontorovich
    Commented Feb 18, 2019 at 21:01

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