2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142758This paper considers some random processes of the form X_{n+1}=TX_n+B_n (mod p) where B_n and X_n are random variables over (Z/pZ)^d and T is a fixed d x d integer matrix which is invertible over the complex numbers. For a particular distribution for B_n, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p)^2 steps suffice to make X_n close to uniformly distributed where X_0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p^b steps are needed for X_n to get close to uniform where b is a value which may depend on T and X_0 is the zero vector.This version incorporates some changes suggested by a referee and is the final pre-publication version. The published version is to appear in the Journal of Theoretical ProbabilityProbability60B15 (primary); 60J10 (secondary)text