parameter error equations in adaptive filtering naturally occur as vector difference equations with random coefficients depending on the realization of the regressors. our aim in this seminar will be to understand the various concepts of and conditions for the exponential stability of the homogeneous parts. specifically, we shall analyze the scalar case in some detail and then proceed to untangle the significantly weaker results available in the vector case. the role of correlation and dependence will be explored to understand the unusual distinction between the cases. please bring your mobile phones, in their off mode, since they will be used as very common example of adaptive filtering and i would like to point to them.