Necessary and sufficient conditions and robustness for convergence of adaptive filtering algorithms

The necessary and sufficient conditions for the almost sure convergence of adaptive filtering algorithms based on {a/n } stepsize are established. Assuming that the covariance of the output signal {Y_n} and the cross-covariance of {Y_n} and the reference signal {ψ_n} are constant, if {Y_n Y_n´} satisfies a law of large numbers (where ´ denotes transpose), then the necessary and sufficient condition for almost sure convergence of adaptive filtering algorithms is that {Y_n ψ_n´} also satisfies the law of large numbers. Moreover when there exists a small deviation from the law of large numbers for { Y_n Y_n´} and {Y_n ψ_n´}, there is also a bounded deviation dependent on the former from the convergence of the algorithms, and the latter tends to zero as the former goes to zero