concentration of functions

Let $X_1,\dots ,X_n$ be random variables in $\mathcal{X}$ and $f:{\mathcal{X}}^n\to [0,\infty )$. What can say about the concentration of $f(X^n)\equiv f(X_1,\dots ,X_n)$?

Of course, one has to impose some assumptions on $f$, otherwise we cannot say anything. Common assumptions include:

• bounded difference assumptions which lead to bounded difference inequalities. This is an assumption on how much changing one coordinate can affect the function value.
• The self-bounding property (see concentration of self-bounding functions). These inequalities typically make use of the Efron-Stein inequality.