The Convergence of Some Recursions

Abstract

In connection with a range of stationary time series models, particularly ARMAX models, recursive calculations of the parameter vector seem important. In these the estimate, $\theta(n)$, from observations to time $n$, is calculated as $\theta(n) = \theta(n - 1) + k_n$ where $k_n$ depends only on $\theta(n - 1), \theta(n - 2), \cdots$ and the data to time $n$. The convergence of two recursions is proved for the simple model $x(n) = \varepsilon(n) + \alpha\varepsilon(n - 1), |\alpha| < 1$, where the $\varepsilon(n)$ are stationary ergodic martingale differences with $E\{\varepsilon(n)^2\mid\mathscr{F}_{n-1}\} = \sigma^2$. The method of proof consists in reducing the study of the recursion to that of a recursion involving the data only through the $\theta(n)$. It seems that many of the recursions introduced for ARMAX models may be treated in this way and the nature of the extensions of the theory is discussed.