Central limits from generating functions

Let (Yₙ) ₙ be a sequence of Rᵈ-valued random variables. Suppose that the generating function (x, z) = ₍ = ₀^ ₘ䂸 (x) zⁿ, \ where ₘ䂸 is the characteristic function of Yₙ, extends to a function on a neighborhood of \0\ \z: |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 such that the distribution of (Yₙ - 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 ₙ Sₙ is a uniformly random permutation, then the distribution of (des (s (ₙ) ) + 1 - (3 - e) n) / n converges, as n, to a normal distribution with variance 2 + 2e - e².