Regularized system identification is the major advance in system identification in the last decade. Although many promising results have been achieved, it is far from complete and there are still many key problems to be solved. One of them is the asymptotic theory, which is about convergence properties of the model estimators as the sample size goes to infinity. The existing related results for regularized system identification are about the almost sure convergence of various hyper-parameter estimators. A common problem of those results is that they do not contain information on the factors that affect the convergence properties of those hyper-parameter estimators, e.g., the regression matrix. In this paper, we tackle problems of this kind for the regularized finite impulse response model estimation with the empirical Bayes (EB) hyper-parameter estimator and filtered white noise input. In order to expose and find those factors, we study the convergence in distribution of the EB hyper-parameter estimator, and the asymptotic distribution of its corresponding model estimator. For illustration, we run Monte Carlo simulations to show the efficacy of our obtained theoretical results.

翻译：正则化系统辨识是过去十年系统辨识领域的重大进展。尽管已取得诸多令人瞩目的成果，但该领域仍有诸多关键问题亟待解决，其中渐近理论是核心问题之一。渐近理论研究当样本量趋于无穷时模型估计量的收敛性质。现有关于正则化系统辨识的相关结论主要集中于各类超参数估计量的几乎必然收敛性。这些结论普遍存在的缺陷在于未能揭示影响超参数估计量收敛特性的因素（如回归矩阵）。本文针对采用经验贝叶斯超参数估计器及滤波白噪声输入的正则化有限脉冲响应模型估计问题展开研究。为揭示并定位这些影响因素，我们深入探究了经验贝叶斯超参数估计量的依分布收敛性，以及其对应模型估计量的渐近分布。通过蒙特卡洛仿真实验，我们验证了所得理论结果的有效性。