Abstract
In this paper, we study the rate of convergence of the Markov chain X n+1=AX n +b n mod p, where A is an integer matrix with nonzero integer eigenvalues and {b n } n is a sequence of independent and identically distributed integer vectors. If λi≠±1 for all eigenvalues λi of A, then n=O((log p)2) steps are sufficient and n=O(log p) steps are necessary to have X n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalue λ1=±1, and λi≠±1 for all i≠1, n=O(p2) steps are necessary and sufficient.
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Asci, C. Generating Uniform Random Vectors. Journal of Theoretical Probability 14, 333–356 (2001). https://doi.org/10.1023/A:1011155412481
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DOI: https://doi.org/10.1023/A:1011155412481