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

Claudio Asci ()
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Claudio Asci: Università degli Studi di Trieste

Journal of Theoretical Probability, 2009, vol. 22, issue 3, 791-809

Abstract: 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.

Keywords: Finite state Markov chains; Fourier transform; Generating random vectors; Rate of convergence; 60B15; 60J10 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0172-8

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