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Generating uniform random vectors in $$\mathbb Z^k_p$$: the general case. (English) Zbl 1228.60078
Summary: We consider the rate of convergence of the Markov chain $${\mathbf X}_{n+1}=A{\mathbf X}_n+\mathbf B_n (\operatorname{mod} p)$$, where $$A$$ is an integer matrix with nonzero eigenvalues, and $$\{{\mathbf B}_n\}_n$$ is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of $${\mathbf Q}^k$$ invariant under $$A$$. If $$|\lambda_i|\neq1$$ for all eigenvalues $$\lambda_i$$ of $$A$$, then $$n=O((\ln p)^2$$) steps are sufficient and $$n=O(\ln p)$$ steps are necessary to have $${\mathbf X}_n$$ sampling from a nearly uniform distribution. Conversely, if $$A$$ has the eigenvalues $$\lambda_i$$ that are roots of positive integer numbers, $$|\lambda_i|=1$$ and $$|\lambda_i|>1$$ for all $$i\neq1$$, then $$O(p^2)$$ steps are necessary and sufficient.

##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F15 Strong limit theorems
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