I'm reading this paper on Pólya urns and I don't understand how to apply a Central Limit Theorem:
Theorem. Assume that $(U(n))_{n\ge0}$ is a Pólya urn of initial composition $(\alpha_1,\dots,\alpha_d)$ and replacement matrix $R=S\cdot Id$ for some $\alpha_1,\dots,\alpha_d$, $S\ge1$. Then, in distribution as $n\uparrow\infty$,$$\frac{U(n)-nSV}{S\sqrt{n}}\xrightarrow{d}\mathcal{N}(0,\Sigma^2)$$, where $\Sigma^2=\sum^d_{i=1}V_ie_i^te_i-V^tV$.
Proof. Previously we've shown that, in distribution, $U(n)=Z(n)$ in distribution, where $Z(n)$ is defined as: conditionally on $V$, $Z(n)=(\alpha_1,\dots,\alpha_d)+S\sum^d_{i=1}X_k$, where $\mathbb{P}(X_k=e_i|V)=V_i$ for all $k\ge1$ and $1\le i\le d$. By the central limit theorem, conditionally on $V$,$$\frac{Z(n)-nSV}{S\sqrt{n}}\xrightarrow{d}\mathcal{N}(0,\Sigma^2)$$ where $\Sigma^2=\text{Cov}(X|V)$, where $X$ is a copy of $X_1$.
Here $V$ is always a $d$-random vector Dirichlet distributed with parameters $(\frac{\alpha_1}S,\dots,\frac{\alpha_d}S)$. I understand that here we do not considere the classical CLT, since we don't have i.i.d. copies of $U(n)$ or $Z(n)$, but they are Pólya urns (so Markov chains). Previously on the paper is shown the almost sure convergence $$\frac{U(n)}{nS}\xrightarrow{n\to\infty}V$$ that surely has to do with this other result, but I don't get how to understand the fluctuations of the random quantity around its almost sure limit. Also, this case in discussed in this other paper, a lot more general, and with a brief parenthesis on the Pólya urn case. I would like to prove the convergence in distribution just on this specific case and why it converges to $\mathcal{N}(0,\text{Cov}(X|V))$