Performance of adaptive estimation algorithms in dependent random environments

We consider the convergence properties of certain algorithms arising in stochastic, discrete-time, adaptive estimation problems and operating in random environments of engineering significance. We demonstrate that the algorithms operating under ideal conditions are describable by homogeneous time-varying linear difference equations with dependent random coefficients, while in practical use, these equations are altered only through the addition of a driving term, accounting for time variation of system parameters, measurement noise, and system undermodeling. We present the concept of almost sure exponential convergence of the homogeneous difference equations as an a priori testable robustness property guaranteeing satisfactory performance in practice. For the three particular algorithms discussed, we present very mild conditions for the satisfaction of this property, and thus explain much of their observed behavior.