Weak convergence and asymptotic properties of adaptive filters with constant gains

The basic adaptive filtering algorithmXₙ₊₁ε = Xₙε - ε Yₙ(Yₙ'Xₙε - ψₙ)is analyzed using the theory of weak convergence. Apart from some very special cases, the analysis is hard when done for each fixedε > 0. But the weak convergence techniques are set up to provide much information for smallε. The relevant facts from the theory are given. Definexε(·)byxε(t) = Xₙεon[nε, nε + ε). Then weak (distributional) convergence of\{xε(·)\}and of\{xε(· + tε)\}is proved under very weak assumptions, wheretε → ε → 0. The normalized errors\{(Xₙε - θ ) / √ε \}are analyzed, where a "stable" point for the "mean" algorithm. The asymptotic properties of a projection algorithm are developed, where theXₙεare truncated at each iteration, if they fall outside of a given set.