Suppose $X_n(z)$ is a sequence of random variables with a boltzmann distribution on $\{1,2,\dots n\}.$ That is $$P(X_n(z)=j)=\frac{c_{j,n} z^j}{F_n(z)}$$ where $F_n(z)=\sum_{j=1}^n c_{j,n}z^j$ is a polynomial known as the partition function.

Suppose uniformly on a compact neighborhood $U$ of $z_0>0$ there is a positive divergent sequence $a_n$ and analytic functions $\ln F(z)$ and $R(z)$ so that $$\ln F_n(z)= a_n \ln F(z)+R(z)+o_U(1)$$ then $\frac{1}{\sqrt{a_n}}X_n(z_0)-\sqrt{a_n}\left(z \frac{d}{dz}\right) \ln F(z)\vert_{z=z_0}\thicksim N(0,\left(z \frac{d}{dz}\right)^2 \ln F(z)\vert_{z=z_0}).$

Given the wealth of mathematical statistical mechanics and probability theory literature on boltzmann distributions is there a theorem similar, or perhaps slightly more general, to this?