Central limits from generating functions

Let (Yₙ)ₙ be a sequence of Rᵈ-valued random variables. Suppose that the generating function \[f(x, z) = ∑n = 0^∞ φY_n(x) z^n,\] where φYₙ is the characteristic function of Yₙ, extends to a function on a neighborhood of \0\ × : |z| ≤ 1\ ⊂ Rᵈ × C which is meromorphic in z and has no zeroes. We prove that if $1 / f(x, z)$ is twice differentiable, then there exists a constant μ such that the distribution of (Yₙ - μ n) / √n converges weakly to a normal distribution as n → ∞. If Yₙ = X₁ + ⋯ + Xₙ, where (Xₙ)ₙ are i.i.d. random variables, then we recover the classical (Lindebergx2013L\'evy) central limit theorem. We also prove the 2020 conjecture of Defant that if πₙ ∈ Sₙ is a uniformly random permutation, then the distribution of (des (s(πₙ)) + 1 - (3 - e) n) / √n converges, as n → ∞, to a normal distribution with variance 2 + 2e - e².