Strong consistency and asymptotic normality of a Gaussian quasi-maximum likelihood estimator for the parameters of a causal, invertible, and identifiable vector autoregressive-moving average (VARMA) model are established in an indirect way. The proof is based on similar results for a much wider class of VARMA models with time-dependent coefficients, hence in the context of non-stationary and heteroscedastic time series. For that reason, the proof avoids spectral analysis arguments and does not make use of ergodicity. The results presented are also applicable to ARMA models.

中文翻译：

---

VARMA 模型估计量渐近性质的间接证明

以间接方式建立因果、可逆和可识别向量自回归移动平均 (VARMA) 模型参数的高斯拟极大似然估计量的强一致性和渐近正态性。该证明基于具有时间相关系数的更广泛类别的 VARMA 模型的类似结果，因此在非平稳和异方差时间序列的背景下。出于这个原因，证明避免了谱分析论证并且没有利用遍历性。呈现的结果也适用于 ARMA 模型。