Central Limit Theorems for Estimators in ARMA Models under Weak Dependence: Theory and Empirical Validation

We establish Central Limit Theorems (CLTs) for Quasi–Maximum Likelihood Estimators (QMLE) in $\text{ARMA}(p,q)$ models under weak dependence, formalized via $\alpha$-mixing. The estimators are $\sqrt{n}$–consistent and asymptotically normal, with covariance in the familiar HAC–sandwich form. To assess finite–sample behavior, we run Monte Carlo experiments for $\text{ARMA}(1,1)$ under Gaussian and non-Gaussian innovations (Student-$t$ with $\nu=3,5$, Gamma with $\alpha=2$, and Poisson with $\lambda=2$). Histograms and Q--Q plots of standardized statistics align closely with $\mathcal N(0,1)$, empirical coverages approach 95\%, and joint scatters of $(\sqrt{n}(\hat\phi-\phi_0), \sqrt{n}(\hat\theta-\theta_0))$ exhibit the expected elliptical shape. The results confirm that QMLE with HAC inference provides reliable estimation and valid uncertainty quantification beyond the classical i.i.d.\ Gaussian setting, offering a practical framework for weakly dependent, non-Gaussian time series.