Inductive methods and rates of Rth-mean convergence in adaptive filtering

Abstract

An inductive method for establishing rates of rth-mean convergence of linear, decreasing-gain adaptive algorithms is introduced. This inductive method does not require any sophisticated probability theory or mathematics and allows such weak forms of dependence as weak-multiplicativity, strong mixing and decaying covariance. In exposing our method, we establish rates of rth-mean convergence for the process , generated by the basic linear adaptive filtering algorithm with and being weakiy-stationary stochastic processes which drive the algorithm and being a sequence of the form k x for some 0 ≥ α ≥ 1. Specifically, we show in most cases that , where r is a real constant greater than or equal to one and . Finally, our method takes advantage of moment bounds for partial sums of non-stationary random sequences, some of which are established within this note