Using Renewal Processes to Generate Long-Range Dependence and High Variability

  • Murad S. Taqqu
  • Joshua B. Levy
Part of the Progress in Probability and Statistics book series (PRPR, volume 11)


We explore here three types of convergence theorems involving the normalized partial sums of two random processes W = W(t) and V = V(t) indexed by the integers t = ...,−1, 0.1,... . W(t) is a stationary renewal reward process with large inter-renewal intervals, while V(t) is a non-stationary process that takes the value zero except at some rare instants t where it achieves extremely high values.


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  1. [1]
    Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Nordhoff.zbMATHGoogle Scholar
  2. [2]
    Levy, J.B. (1983). High variability and long-range dependence: a modeling approach based on renewal sequences. M.S. thesis: Cornell University.Google Scholar
  3. [3]
    Mandelbrot, B.B. (1969). Long-run linearity, locally Gaussian processes, H-spectra, and infinite variances. International Economic Review, 10, 82–113.CrossRefzbMATHGoogle Scholar
  4. [4]
    Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. San Francisco: W.H. Freeman Co.zbMATHGoogle Scholar
  5. [5]
    Mandelbrot, B.B. and Taqqu, M.S. (1979). Robust R/S analysis of long-run serial correlation. Proceedings of the 42nd Session of the International Statistical Institute, Manila. Bulletin of the I.S.I., Vol. 48, Book 2, pp. 69–104.MathSciNetGoogle Scholar
  6. [6]
    Mijnheer, J.L. (1975). Sample Path Properties of Stable Processes. Mathematical Centre Tracts 59. Amsterdam: Mathematisch Centrum.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Murad S. Taqqu
    • 1
  • Joshua B. Levy
    • 2
  1. 1.Department of MathematicsBoston UniversityBostonUSA
  2. 2.School of BusinessState University of New YorkAlbanyUSA

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