Asymptotic behavior of an affine random recursion in (Z_p)^k defined by a matrix with an eigenvalue of size 1

In this paper we study the rate of convergence of the Markov chain XnC1 D AXn C Bn.mod p/, where A is an integer invertible matrix, and fBngn is a sequence of independent and identically distributed integer vectors. If A has an eigenvalue of size 1, then n D O.p2/ steps are necessary and sufficient to have Xn sampling from a nearly uniform distribution.