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Asymptotic behavior of an affine random recursion in $$\mathbf Z_p^k$$ defined by a matrix with an eigenvalue of size 1. (English) Zbl 1171.65007
Summary: We study the rate of convergence of the Markov chain $$\mathbf X_{n+1}=A\mathbf X_n+\mathbf B _n$$(mod $$p$$), where $$A$$ is an integer invertible matrix, and $$\{\mathbf B_n\}$$(mod $$p$$) is a sequence of independent and identically distributed integer vectors. If $$A$$ has an eigenvalue of size 1, then $$n=O(p^{2})$$ steps are necessary and sufficient to have $$\mathbf X_n$$ sampling from a nearly uniform distribution.

MSC:
 65C40 Numerical analysis or methods applied to Markov chains 60J22 Computational methods in Markov chains
Keywords:
convergence; Markov chain
Full Text:
References:
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