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 $f:\mathbb R^N \rightarrow \mathbb R$ be a 1-Lipschitz function. Is it possible to establish a concentration inequality of the type:

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

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:

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

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 $f:\mathbb R^N \rightarrow \mathbb R$ be a 1-Lipschitz function. Is it possible to establish a concentration inequality of the type:

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

Should it be $c\propto N$?

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 $f:\mathbb R^N \rightarrow \mathbb R$ be a 1-Lipschitz function. Is it possible to establish a concentration inequality of the type:

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