Peak Effects in Stable Linear Difference Equations

TL;DR

This paper investigates the peak deviations in stable linear difference equations, providing exact bounds and analyzing the effects of initial conditions and noise, with implications for understanding transient behaviors before convergence.

Contribution

It offers new insights into the finite-time peak deviations in stable difference equations, including exact bounds and analysis of non-random noise effects.

Findings

Solutions can deviate significantly before converging to zero.

Exact deviation values and lower bounds are derived for various root distributions.

Upper bounds are established for difference equations with bounded noise.

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 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.