Lyapunov Techniques for the Exponential Stability of Linear Difference Equations with Random Coefficients

We consider an approach to studying the exponential stability of linear difference equations with random coefficients through the use of Lyapunov stability techniques. The equations we study are of a form familiar from adaptive estimation algorithms, which motivates the examination. It is necessary to define the almost sure exponential convergence of a random process, and then to derive sufficient conditions on the coefficients of the difference equations to ensure the almost sure exponential convergence of the state. We consider, in particular, two very reasonable types of random coefficients—ergodic and stationary and Ø-mixing and nonstationary—which would appear to encompass many engineering situations. An example of the power of the theory is given, where it is applied to a common adaptive filtering algorithm to derive mild conditions for exponential convergence with dependent random inputs.