  We consider asymptotically stable scalar difference equations with unit-norm
initial conditions. First, it is shown that the solution may happen to deviate
far away from the equilibrium point at finite time instants prior to converging
to zero. Second, for a number of root distributions and initial conditions,
exact values of deviations or lower bounds are provided. Several specific
difference equations known from the literature are also analyzed and estimates
of deviations are proposed. Third, we consider difference equations with
non-random noise (i.e., bounded-noise autoregression) and provide upper bounds
on the solutions. Possible generalizations, e.g., to the vector case are
discussed and directions for future research are outlined.
