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Generating Uniform Random Vectors

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  • Claudio Asci

    (Università degli Studi di L'Aquila)

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.

Suggested Citation

  • Claudio Asci, 2001. "Generating Uniform Random Vectors," Journal of Theoretical Probability, Springer, vol. 14(2), pages 333-356, April.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:2:d:10.1023_a:1011155412481
    DOI: 10.1023/A:1011155412481
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    References listed on IDEAS

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    1. Baxter, J. R. & Rosenthal, Jeffrey S., 1995. "Rates of convergence for everywhere-positive Markov chains," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 333-338, March.
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    Cited by:

    1. Claudio Asci, 2009. "Generating Uniform Random Vectors in Z p k : The General Case," Journal of Theoretical Probability, Springer, vol. 22(3), pages 791-809, September.
    2. Martin Hildebrand & Joseph McCollum, 2008. "Generating Random Vectors in (ℤ/pℤ) d via an Affine Random Process," Journal of Theoretical Probability, Springer, vol. 21(4), pages 802-811, December.

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