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.