Convergence analysis of a linear recursive algorithm for arma processes

A recursive version of a three stage linear estimation scheme for ARMA processes has recently been proposed and tested. In the first stage an autoregressive model is fitted. In the second stage an ARMA model is fitted using the residuals obtained in the first stage. The data (output, input and residuals) are then filtered using the model obtained from the second stage and an improved ARMA model is fitted to the filtered data. In this note the almost sure convergence of the algorithm is established using a relatively simple procedure.