Peak effects in stable linear difference equations

ABSTRACT

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 analysed and estimates of deviations are proposed. Third, we consider difference equations with non-random noise (ie bounded-noise autoregression) and provide upper bounds on the solutions. Possible generalizations, eg to the vector case are discussed and directions for future research are outlined.

KEYWORDS:

• Linear difference equations
• stability
• nonzero initial conditions
• peak effect
• large deviations
• non-random noise

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first two authors was supported by the Russian Science Foundation through grant no. 16-11-10015.