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Generating Random Vectors in (ℤ/pℤ)d via an Affine Random Process

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Abstract

This paper considers some random processes of the form X n+1=T X n +B n (mod p) where B n and X n are random variables over (ℤ/pℤ)d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)2 steps suffice to make X n close to uniformly distributed where X 0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p b steps are needed for X n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X 0 is the zero vector.

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References

  1. Asci, C.: Generating uniform random vectors. J. Theor. Probab. 14, 333–356 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chung, F., Diaconis, P., Graham, R.: A random walk problem arising in random number generation. Ann. Probab. 15, 1148–1165 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Diaconis, P.: Group Representations in Probability and Statistics. Institute of Mathematical Statistics, Hayward (1988)

    MATH  Google Scholar 

  4. Greenhalgh, A.: Random walks on groups with subgroup invariance properties. Ph.D. thesis, Stanford University, Department of Mathematics (1989)

  5. Hildebrand, M.: Rates of convergence of some random processes on finite groups. Ph.D. thesis, Harvard University, Department of Mathematics (1990)

  6. Hildebrand, M.: Random processes of the form X n+1=a n X n +b n (mod p). Ann. Probab. 21, 710–720 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hildebrand, M.: Random processes of the form X n+1=a n X n +b n (mod p) where b n takes on a single value. In: Aldous, Pemantle (eds.) Random Discrete Structures, pp. 153–174. Springer, New York (1996)

    Google Scholar 

  8. Knuth, D.: The Art of Computer Programming, vol. II, 2nd edn. Addison-Wesley, Menlo Park (1981)

    Google Scholar 

  9. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications, revised edn. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY, 12222, USA

    Martin Hildebrand

  2. Department of Mathematical Sciences, Elms College, Chicopee, MA, 01013, USA

    Joseph McCollum

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  1. Martin Hildebrand
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  2. Joseph McCollum
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Correspondence to Martin Hildebrand.

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Hildebrand, M., McCollum, J. Generating Random Vectors in (ℤ/pℤ)d via an Affine Random Process. J Theor Probab 21, 802–811 (2008). https://doi.org/10.1007/s10959-007-0135-5

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  • DOI: https://doi.org/10.1007/s10959-007-0135-5

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