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Generating Uniform Random Vectors in Z p k : The General Case

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  • Claudio Asci

    (Università degli Studi di Trieste)

Abstract

We consider the rate of convergence of the Markov chain X n+1=A X n +B n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B n } n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q k invariant under A. If $|\lambda_{i}|\not =1$ for all eigenvalues λ i of A, then n=O((ln p)2) steps are sufficient and n=O(ln p) steps are necessary to have X n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ i that are roots of positive integer numbers, |λ 1|=1 and |λ i |>1 for all $i\not =1$ , then O(p 2) steps are necessary and sufficient.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-008-0172-8
    DOI: 10.1007/s10959-008-0172-8
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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.
    2. Claudio Asci, 2001. "Generating Uniform Random Vectors," Journal of Theoretical Probability, Springer, vol. 14(2), pages 333-356, April.
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