Let $P$ be a probability measure and $f$ be some probability density function (not necessarily related to $P$). Consider the function $$ L(X_1,\ldots,X_n) =\frac1n\sum_{i=1}^n\log f(X_i), \quad X_i\overset{iid}{\sim}P. $$

This is just the empirical log-likelihood from statistics. Are there any concentration inequalities that are specialized for $L(X_1,\ldots,X_n)$?

Surprisingly, I could not find any in the literature besides the generic ones. That is, we can naively apply generic inequalities such as Bernstein, etc. or assume that $\log f(X_i)$ is a subgaussian random variable, but this is a bit unsatisfying. For example, in the simplest case where $f$ is a Gaussian density, $\log f(x)$ is a quadratic function, and hence not subgaussian if $X_i$ is also Gaussian.

It seems like it should be possible to exploit the fact that $f$ is a density to come up with something more useful.