The2rth mean convergence of adaptive filters with stationary dependent random variables

The
2r
th mean convergence of an adaptive filtering algorithm
C_{n+ 1} = C_{n} - a_{n}(C_{n}^{T} X_{n} - \alpha_{n}) X_{n}
is studied, where
\alpha_{n}
's and
X_{n}
's are useful random signals and observation vectors, respectively, and
\{ a_{n} \}
is a sequence of positive numbers decreasing to zero. In this work, we suppose that the sequence
\{(\alpha_{n},X_{n})\}
is a stationary mixing sequence (
\rho
-mixing,
\phi
-mixing, or
\psi
-mixing). Under some additional conditions it is shown that
C_{n}
converges to
c_{\ast}
in the
2r
th mean, where
c_{\ast}
is the unique solution of the Wiener-Hopf equation.