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.

从文献中可知，齐次线性稳定差分方程的解可能会在有限时刻与非零初始条件出现较大的偏差或峰值。虽然从确定性的角度研究了这个问题，但当初始条件和方程的系数都具有随机性质时，对于此类事件发生的概率知之甚少。在本文中，通过利用 Schur 域体积的结果，我们能够计算发生偏差的概率。事实证明，这非常接近于一，即使对于低度方程也是如此。因此，我们声称“稳定差分方程的解可能经历峰值”. 然后，我们利用统计学习的工具来解决其他问题，例如评估峰值的平均幅度和最大值。