Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with law $\mu$. The Sanov Theorem then states that the empirical measures $$ \mu^N =\frac{1}{N} \sum _{n=1}^N\delta _{X_n} $$ satisfy a large deviation principle at speed $N$ with good rate function $H(.\mid\mu)$, $H$ being the relative entropy.

I was wondering, what is known if we consider a sequence of independent random variables $(X_n)_{n\in\mathbb N}$ but with different laws $(\mu_n)_{n\in\mathbb N}$ ? For example, what if you take $X_n=Y_n+a_n$, where $(Y_n)_{n\in\mathbb n}$ is a sequence of i.i.d random variables with law $\mu$, and $(a_n)_{n\in\mathbb N}$ is a sequence of real numbers such that $$ \frac{1}{N} \sum _{n=1}^N\delta _{a_n}\rightarrow \nu \qquad \mbox{(weakly)} $$ for some probability measure $\nu$ as $N\rightarrow \infty$ ?

EDIT : (after the comment of Anthony Quas) Let's say we may assume the that the convergence rate for the $a_n$'s is as you want, for example at the rate $\exp(-N.)$. My interest is more about what would be the rate function.