Central limits from generating functions

Let $(Y_n{)}_n$ be a sequence of ${\mathbb{R}}^d$-valued random variables. Suppose that the generating function $$f(x,z)=\sum _{n=0}^{\mathrm{\infty }}{\phi }_{Y_n}(x)z^n,$$ where ${\phi }_{Y_n}$ is the characteristic function of $Y_n$, extends to a function on a neighborhood of $\{0\}\times \{z:\mathrm{\mid }z\mathrm{\mid }\le 1\}\subset {\mathbb{R}}^d\times \mathbb{C}$ which is meromorphic in $z$ and has no zeroes. We prove that if $1\mathrm{/}f(x,z)$ is twice differentiable, then there exists a constant $\mu$ such that the distribution of $(Y_n-\mu n)\mathrm{/}\sqrt{n}$ converges weakly to a normal distribution as $n\to \mathrm{\infty }$. If $Y_n=X_1+\cdots +X_n$, where $(X_n{)}_n$ are i.i.d. random variables, then we recover the classical (Lindeberg$\text{unicode}x2013$Lévy) central limit theorem. We also prove the 2020 conjecture of Defant that if ${\pi }_n\in {\mathfrak{S}}_n$ is a uniformly random permutation, then the distribution of $(\mathrm{des}(s({\pi }_n))+1-(3-e)n)\mathrm{/}\sqrt{n}$ converges, as $n\to \mathrm{\infty }$, to a normal distribution with variance $2+2e-e^2$.