Generating uniform random vectors in Zp: the general case

This paper is about the rate of convergence of the Markov chain Xn+1 = AXn+Bn (mod p), where A is an integer matrix with nonzero eigenvalues and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q invariant under A. If |λi| 6 = 1 for all eigenvalues λi of A, then n = O ( (ln p) ) steps are sufficient and n = O(ln p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λi that are roots of positive integer numbers, |λ1| = 1 and |λi| > 1 for all i 6= 1, then O ( p ) steps are necessary and sufficient. Running head. Generating uniform random vectors.1 MSC 2000 subject classifications. Primary 60B15; secondary 60J10.