On the Gaussian Limit of the Output of IIR Filters

Abstract

We study the asymptotic distribution of the output of a stable Linear Time-Invariant (LTI) system driven by a non-Gaussian stochastic input. Motivated by longstanding heuristics in the stochastic describing function method, we rigorously characterize when the output process becomes approximately Gaussian, even when the input is not. Using the Wasserstein-1 distance as a quantitative measure of non-Gaussianity, we derive upper bounds on the distance between the appropriately scaled output and a standard normal distribution. These bounds are obtained via Stein's method and depend explicitly on the system's impulse response and the dependence structure of the input process. We show that when the dominant pole of the system approaches the edge of stability and the input satisfies one of the following conditions: (i) independence, (ii) positive correlation with a real and positive dominant pole, or (iii) sufficient correlation decay, the output converges to a standard normal distribution at rate $O(1/\sqrt{t})$. We also present counterexamples where convergence fails, thereby motivating the stated assumptions. Our results provide a rigorous foundation for the widespread observation that outputs of low-pass LTI systems tend to be approximately Gaussian.