On the Gaussian Limit of the Output of IIR Filters

TL;DR

This paper rigorously analyzes when the output of stable LTI systems driven by non-Gaussian inputs becomes approximately Gaussian, providing bounds and conditions for convergence to normality.

Contribution

It offers the first rigorous characterization of Gaussian convergence for LTI outputs driven by non-Gaussian inputs, using Stein's method and explicit bounds.

Findings

Output converges to Gaussian when the dominant pole nears stability edge and input has certain dependence properties.

Explicit bounds on non-Gaussianity are derived using Wasserstein-1 distance.

Counterexamples show convergence can fail under certain conditions.

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.