Weak convergence and asymptotic properties of adaptive filters with constant gains

The basic adaptive filtering algorithm
X_{n+1}^{\epsilon} = X_{n}^{\epsilon} - \epsilon Y_{n}(Y_{n}^{'}X_{n}^{\epsilon} - \psi_{n})
is analyzed using the theory of weak convergence. Apart from some very special cases, the analysis is hard when done for each fixed
\epsilon > 0
. But the weak convergence techniques are set up to provide much information for small
\epsilon
. The relevant facts from the theory are given. Define
x^{\epsilon}(\cdot)
by
x^{\epsilon}(t) = X_{n}^{\epsilon}
on
[n\epsilon, n\epsilon + \epsilon)
. Then weak (distributional) convergence of
\{x^{\epsilon}(\cdot)\}
and of
\{x^{\epsilon}(\cdot + t_{\epsilon})\}
is proved under very weak assumptions, where
t_{\epsilon} \rightarrow \infty
as
\epsilon \rightarrow 0
. The normalized errors
\{(X_{n}^{\epsilon} - \theta ) / \sqrt{\epsilon} \}
are analyzed, where
\theta
is a "stable" point for the "mean" algorithm. The asymptotic properties of a projection algorithm are developed, where the
X_{n}^{\epsilon}
are truncated at each iteration, if they fall outside of a given set.