A Probabilistic Point of View on Peak Effects in Linear Difference Equations

Abstract It is known from the literature that solutions of homogeneous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. While the problem has been studied from a deterministic standpoint, not much is known about the probability of occurrence of such event when both the initial conditions and the coefficients of the equation have random nature. In this paper, by exploiting results on the volume of the Schur domain, we are able to compute the probability for deviations to occur. This turns out to be very close to unity, even for equations of low degree. Hence, we claim that “solutions of stable difference equations probably experience peak”. Then, we make use of tools from statistical learning to address other issues such as evaluation of the mean magnitude and maximum value of peak.