On the almost sure central limit theorem for ARX processes in adaptive tracking

Abstract

The goal of this paper is to highlight the almost sure central limit theorem (ASCLT) for martingales to the control community. We shall establish the ASCLT for the least squares estimator of the unknown parameter of a controllable ARX(p, q) process in adaptive tracking. The usual notion of controllability for ARX(p, q) processes allows us to avoid the use of a persistent excitation in the adaptive tracking control. We shall also provide strongly consistent estimators of the even moments of the driven noise as well as two quadratic strong laws for the average costs and the estimation errors sequences. Our theoretical results are illustrated by numerical experiments.

On the almost sure central limit theorem for the elephant random walk

“…In the context of the ASCLT for scalar martingales we may refer to the reader to the works of Bercu [3], Bercu and Fort [6] and to the important contributions of Chaâbane [14], Chaâbane and Maâouia [15], Chaâbane and Touati [16] and Lifshits [27,28]. In addition, The ASCLT has been widely investigated in stochastic approximation theory [5,12,32], in statistics [10,23,31] and recently in ARX adaptive tracking [8].…”