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Generating random vectors in $$(\mathbb Z/ p \mathbb Z)^d$$ via an affine random process. (English) Zbl 1159.65004
The work improves some results of C. Asci [J. Theor. Probab. 14, No. 2, 333–356 (2001; Zbl 1005.65007)], and continues with previous works of the first author; for example, see M. Hildebrand [Ann. Probab. 21, No. 2, 710–720 (1993; Zbl 0776.60012)]. The authors consider the random processes $$\mathbf X_{n+1}=T\mathbf X_n+\mathbf B_n\pmod p$$ where $$\mathbf B_n$$ and $$\mathbf X_n$$ are random variables over $$(\mathbb Z/p\mathbb Z)^d$$ and $$T$$ is a fixed $$d\times d$$ integer matrix which is invertible over $$\mathbb C$$. If $$T$$ has no eigenvalues of modulus $$1$$ over $$\mathbb C$$, sufficient conditions are given to make $$\mathbf X_n$$ close to uniformly distributed. In case $$T$$ has a complex eigenvalue which is a root of unity, necessary conditions are given.

##### MSC:
 65C10 Random number generation in numerical analysis 60G07 General theory of stochastic processes
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##### References:
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