Elementwise decoupling and convergence of the Riccati equation in the SG algorithm

It is shown that the difference Riccati equation of the Stenlund–Gustafsson (SG) algorithm for estimation of linear regression models can be solved elementwise. Convergence estimates for the elements of the solution to the Riccati equation are provided, directly relating convergence rate to the signal-to-noise ratio in the regression model. It is demonstrated that the elements of the solution lying in the direction of excitation exponentially converge to a stationary point while the other elements experience bounded excursions around their current values.