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The 2r th mean convergence of adaptive filters with stationary dependent random variables

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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.

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Information Theory, IEEE Transactions on  (Volume:30 ,  Issue: 2 )