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Complexity Questions in Non-Uniform Random Variate Generation

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Abstract

In this short note, we recall the main developments in non-uniform random variate generation, and list some of the challenges ahead.

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References

  • AHRENS, J.H. and DIETER, U. (1974): Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing, vol. 12, pp. 223–246.

    Article  MathSciNet  MATH  Google Scholar 

  • AKHIEZER, N.I. (1965): The Classical Moment Problem, Hafner, New York.

    MATH  Google Scholar 

  • ALSMEYER, G. and IKSANOV, A. (2009): A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks.’ Electronic Journal of Probability, vol. 14, pp. 289–313.

    MathSciNet  MATH  Google Scholar 

  • ASMUSSEN, S., GLYNN, P. and THORISSON, H. (1992): Stationary detection in the initial transient problem. ACM Transactions on Modeling and Computer Simulation, vol. 2, pp. 130–157.

    Article  MATH  Google Scholar 

  • BAILEY, R.W. (1994): Polar generation of random variates with the t distribution (1994): Mathematics of Computation, vol. 62, pp. 779–781.

    MathSciNet  MATH  Google Scholar 

  • BONDESSON, L. (1982): On simulation from infinitely divisible distributions. Advances in Applied Probability, vol. 14, pp. 855–869.

    Article  MathSciNet  MATH  Google Scholar 

  • BOX, G.E.P. and MÜLLER, M.E. (1958): A note on the generation of random normal deviates. Annals of Mathematical Statistics, vol. 29, pp. 610–611.

    Article  MATH  Google Scholar 

  • CHAMBERS J.M., MALLOWS, C.L. and STUCK, B.W. (1976): A method for simulating stable random variables. Journal of the American Statistical Association, vol. 71, pp. 340–344.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (1981a): The series method in random variate generation and its application to the Kolmogorov-Smirnov distribution. American Journal of Mathematical and Management Sciences, vol. 1, pp. 359–379.

    MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (1981b): The computer generation of random variables with a given characteristic function. Computers and Mathematics with Applications, vol. 7, pp. 547–552.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (1986a): Non-Uniform Random Variate Generation, Springer-Verlag, New York.

    MATH  Google Scholar 

  • DEVROYE, L. (1986b): An automatic method for generating random variables with a given characteristic function. SIAM Journal of Applied Mathematics, vol. 46, pp. 698–719.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (1989): On random variate generation when only moments or Fourier coefficients are known. Mathematics and Computers in Simulation, vol. 31, pp. 71–89.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (1991): Algorithms for generating discrete random variables with a given generating function or a given moment sequence. SIAM Journal on Scientific and Statistical Computing, vol. 12, pp. 107–126.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (1996): Random variate generation in one line of code. In: 1996 Winter Simulation Conference Proceedings, Charnes, J.M., Morrice, D.J., Brunner D.T. and Swain J.J. (eds.), pp. 265–272, ACM, San Diego, CA.

    Google Scholar 

  • DEVROYE, L. (1997): Simulating theta random variates. Statistics and Probability Letters, vol. 31, pp. 2785–2791.

    Article  MathSciNet  Google Scholar 

  • DEVROYE, L., FILL, J., and NEININGER, R. (2000): Perfect simulation from the quicksort limit distribution. Electronic Communications in Probability, vol. 5, pp. 95–99.

    MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (2001): Simulating perpetuities. Methodologies and Computing in Applied Probability, vol. 3, pp. 97–115.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. and NEININGER, R. (2002): Density approximation and exact simulation of random variables that are solutions of fixed-point equations. Advances of Applied Probability, vol. 34, pp. 441–468.

    Article  MathSciNet  MATH  Google Scholar 

  • DEVROYE, L. (2009): On exact simulation algorithms for some distributions related to Jacobi theta functions. Statistics and Probability Letters, vol. 21, pp. 2251–2259.

    Article  MathSciNet  Google Scholar 

  • DEVROYE, L. and FAWZI, O. (2010): Simulating the Dickman distribution. Statistics and Probability Letters, vol. 80, pp. 242–247.

    Article  MathSciNet  MATH  Google Scholar 

  • FILL, J. (1998): An interruptible algorithm for perfect sampling via Markov chains. The Annals of Applied Probability, vol. 8, pp. 131–162.

    Article  MathSciNet  MATH  Google Scholar 

  • FILL, J.A. and HUBER, M (2009): Perfect simulation of perpetuities, To appear.

    Google Scholar 

  • FLAJOLET, P. and SAHEB, N. (1986): The complexity of generating an exponentially distributed variate. Journal of Algorithms, vol. 7, pp. 463–488.

    Article  MathSciNet  MATH  Google Scholar 

  • GOLDIE, C.M. and MALLER, R.A. (2000): Stability of perpetuities. Annals of Probability, vol. 28, pp. 1195–1218.

    Article  MathSciNet  MATH  Google Scholar 

  • GREEN, P.J. and MURDOCH, D.J. (2000): Exact sampling for Bayesian inference: towards general purpose algorithms (with discussion). In: Monte Carlo Methods, Bernardo, J.M., Berger, J.O., Dawid, A.P. and Smith, A.F.M. (eds.), pp. 301–321, Bayesian Statistics, vol. 6, Oxford university Press, Oxford.

    Google Scholar 

  • HASTINGS, C. (1955): Approximations for Digital Computers, Princeton University Press, Princeton, New Jersey.

    MATH  Google Scholar 

  • HÖRMANN, W., LEYDOLD, J., and DERFLINGER, G. (2004): Automatic Nonuniform Random Variate Generation, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • HUFFMAN, D. (1952): A method for the construction of minimum-redundancy codes. Proceedings of the IRE, vol. 40, pp. 1098–1101.

    Article  Google Scholar 

  • KANTER, M. (1975): Stable densities under change of scale and total variation inequalities. Annals of Probability, vol. 3, pp. 697–707.

    Article  MathSciNet  MATH  Google Scholar 

  • KEANE, M.S., and O’BRIEN, G.L. (1994): A Bernoulli factory. ACM Transactions on Modeling and Computer Simulation, vol. 4, pp. 213–219.

    Article  MATH  Google Scholar 

  • KENDALL, W. (2004): Random walk CFTP. Thönnes ed., Department of Statistics, University of Warwick.

    Google Scholar 

  • KNUTH, D.E. and YAO, A.C. (1976): The complexity of nonuniform random number generation. in: Algorithms and Complexity, Traub, J.E. (ed.), pp. 357–428, Academic Press, New York, N.Y..

    Google Scholar 

  • MARSAGLIA, G. (1968): Random numbers fall mainly in the planes. Proceedings of the National Academy of Sciences, vol. 60, pp. 25–28.

    Article  MathSciNet  Google Scholar 

  • MARSAGLIA, G. and ZAMAN, A. (1991): A new class of random number generators. Annals of Applied Probability, vol. 1, pp. 462–480.

    Article  MathSciNet  MATH  Google Scholar 

  • METROPOLIS, N., ROSENBLUTH, A., ROSENBLUTH, M., TELLER, A., and TELLER, E. (1953): Equations of state calculations by fast computing machines. Journal of Chemical Physics, vol. 21, p. 1087–1091.

    Article  Google Scholar 

  • MURDOCH, D.J. and GREEN, P.J. (1998): Exact sampling from a continous space. Scandinavian Journal of Statistics, vol. 25, pp. 483–502.

    Article  MathSciNet  MATH  Google Scholar 

  • PROPP, G.J. and WILSON, D.B. (1996): Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms, vol. 9, pp. 223–252.

    Article  MathSciNet  MATH  Google Scholar 

  • RÖSLER, U. and RÜSHENDORF, L. (2001): The contraction method for recursive algorithms. Algorithmica, vol. 29, pp. 3–33.

    Article  MathSciNet  MATH  Google Scholar 

  • K. SATO (2000): Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.

    Google Scholar 

  • ULRICH, U. (1984): Computer generation of distributions on the m-sphere. Applied Statistics, vol. 33, pp. 158–163.

    Article  MathSciNet  MATH  Google Scholar 

  • VERVAAT, W. (1979): On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Advances in Applied Probability, vol. 11, pp. 750–783.

    Article  MathSciNet  MATH  Google Scholar 

  • VON NEUMANN, J. (1963): Various techniques used in connection with random digits. Collected Works, vol. 5, pp. 768–770, Pergamon Press. Also in (1951): Monte Carlo Method. National Bureau of Standards Series, Vol. 12, pp. 36-38.

    Google Scholar 

  • WILSON, D.B. (2000): Layered multishift coupling for use in perfect sampling algorithms (with a primer on CFTP). In: Monte Carlo Methods, Madras, N. (ed.), pp. 141–176, Fields Institute Communications, vol. 6, American Mathematical Society.

    Google Scholar 

  • ZOLOTAREV, V. M. (1959): On analytic properties of stable distribution laws. Selected Translations in Mathematical Statistics and Probability, vol. 1, pp. 207–211.

    Google Scholar 

  • ZOLOTAREV, V. M. (1966): On the representation of stable laws by integrals. Selected Translations in Mathematical Statistics and Probability, vol. 6, pp. 84–88.

    Google Scholar 

  • ZOLOTAREV, V. M. (1981): Integral transformations of distributions and estimates of parameters of multidimensional spherically symmetric stable laws. In: Contributions to Probability, pp. 283–305, Academic Press.

    Google Scholar 

  • ZOLOTAREV, V. M. (1986): One-Dimensional Stable Distributions, American Mathematical Society, Providence, R.I..

    Google Scholar 

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Correspondence to Luc Devroye .

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Devroye, L. (2010). Complexity Questions in Non-Uniform Random Variate Generation. In: Lechevallier, Y., Saporta, G. (eds) Proceedings of COMPSTAT'2010. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2604-3_1

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