Generation of some First-Order Autoregressive Markovian Sequences of Positive Random Variables with Given Marginal Distributions

  • A. J. Lawrance
  • P. A. W. Lewis
Part of the Progress in Computer Science book series (PCS, volume 2)


Methods for simulating dependent sequences of continuous positivevalued random variables with exponential, uniform, Gamma, and mixed exponential marginal distributions are given. In most cases the sequences are first-order, linear autoregressive, Markovian processes. A very broad two-parameter family of this type, GNEAR(l), with exponential marginals and both positive and negative correlation is defined and its transformation to a similar multiplicative process with uniform marginals is given. It is shown that for a subclass of this two-parameter family extension to mixed exponential marginals is possible, giving a model of broad applicability for analyzing data and modelling stochastic systems, although negative correlation is more difficult to obtain than in the exponential case. Finally, two schemes for autoregressive sequences with Gamma distributed marginals are outlined. Efficient simulation of some of these schemes is discussed.


Sample Path Dependent Sequence Exponential Random Variable Uniform Random Number Positive Random Variable 
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  1. [1]
    Gaver, D. P. and Lewis, P. A. W., “First Order Autoregressive Gamma Sequences and Point Processes,” Adv. in Appl. Probab., 12, 1980, 727–745.Google Scholar
  2. [2]
    Lawrance, A. J. and Lewis, P. A. W., “A Moving Average Exponential Point Process (EMA1),” J. Appl. Probab., 14, 1977, 98–113.CrossRefGoogle Scholar
  3. [3]
    Jacobs, P. A. and Lewis, P. A. W., “A Mixed Autoregressive-Moving Average Exponential sequence and Point Process (EARMA 1,1),” Adv. in Appl. Probab., 9, 1977, 87–104.Google Scholar
  4. [4] Lawrance, A. J. and Lewis, P. A. W., The Exponential Autoregressive Moving Average EARMA(p,q)
    Process J. Roy. Statist. Soc. Ser. B, 42, 1980, 150–161.Google Scholar
  5. [5]
    Lawrance, A. J. and Lewis, P. A. W., “A New Autoregressive Time-Series Model in Exponential Variables (NEAR(l)),” Adv. In Appl. Probab., 13, 1981, 826–845.Google Scholar
  6. [6]
    Lawrance, A. J., “The Mixed Exponential Solution to the First-Order Autoregressive Model,” J. Appl. Probab., 17, 1980, 546–552.CrossRefGoogle Scholar
  7. [7]
    Lawrance, A. J., “Some Autoregressive Models for Point Processes,” Point Process and Queuing Problems, Colloquia Mathematica Societates Janos Bolyai, 24, P. Bartfai and J. Tomko eds., North Holland, 1981, Amsterdam, 257–275.Google Scholar
  8. [8]
    Lewis, P. A. W., “Simple Models for Positive-Valued and Discrete-Valued Time Series with ARMA Correlation Structure,” Proc. Fifth International Symposium Mult. Anal., Multivariate Analysis-V, P. R. Krishnaiah, ed. North-Ho Hand, Amsterdam, 1980, 151–166.Google Scholar
  9. [9]
    Lawrance, A. J. and Lewis, P. A. W., “Simulation of some Autoregressive sequences of positive random variables.” Proc. 1979 Winter Simulation Conference, Highland, H.J., Spiegel, M. G. and Shannon, R. eds., 1979. IEEE Press, N.Y., 301–308.Google Scholar
  10. [10]
    Cox, D. R. and Lewis, P. A. W., Statistical Analysis of Series of Events, Methuen, London, 1966.Google Scholar
  11. [11]
    Lewis, P. A. W. and Shedler, G. S., “Analysis and Modelling of Point Processes in Computer Systems,” Bull. ISI, XLVII (2), 1978, 193–219.Google Scholar
  12. [12]
    Jacobs, P. A., “A Closed Cyclic Queueing Network with Dependent Exponential Service Times,” J. Probab., 15, 1978, 573–589.CrossRefGoogle Scholar
  13. [13]
    Jacobs, P. A., “Heavy traffic results for single-server queues with dependent (EARMA) service and interarrival times,” Adv. in Appl. Probab., 12, 1980, 517–529.Google Scholar
  14. [14]
    Jacobs, P. A. and Lewis, P. A. W., “Discrete Time Series Generated by Mixtures, I: Correlational and Runs Properties,” J. Roy. Statist. Soc. Ser. B., 40, 1978, 94–105.Google Scholar
  15. [15]
    Jacobs, P. A. and Lewis, P. A. W., “Discrete Time Series Generated by Mixtures II: Asymptotic Properties,” J. Roy. Statist. Soc. Ser. B, 40, 1978, 222–228.Google Scholar
  16. [16]
    Vervaat, W., “On a stochastic difference equation and a representation of non-negative infinitely divisible random variables,” Adv. in Appl. Probab., 11, 1979, 750–783.Google Scholar
  17. [17]
    G. S. Fishman, personal communication.Google Scholar
  18. [18]
    Lawrance, A. J. and Lewis, P. A. W., “A mixed exponential time series model, NMEAR(p,q),” Naval Postgraduate School Technical Report NPS55-80-012, 1980. To appear in Management Sci.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

  • A. J. Lawrance
    • 1
  • P. A. W. Lewis
    • 2
  1. 1.Dept. StatisticsUniversity of BirminghamBirminghamEngland
  2. 2.Dept. Operations ResearchNaval Postgraduate SchoolMontereyUSA

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