Summary: This paper analyzes stochastic linear discrete-time processes, whose process noise sequence consists of independent and uniformly distributed random variables on given zonotopes. We propose a cumulant-based approach for approximating both the transient and limit distributions of the associated state sequence. The method relies on a novel class of \(k\)-symmetric Lyapunov equations, which are used to construct explicit expressions for the cumulants. The state distribution is recovered via a generalized Gram-Charlier expansion with respect to products of a multivariate variant of Wigner’s semicircle distribution using Chebyshev polynomials of the second kind. This expansion converges uniformly, under surprisingly mild conditions, to the exact state distribution of the system. A robust feedback control synthesis problem is used to illustrate the proposed approach.