• Bernoulli
  • Volume 15, Number 1 (2009), 195-222.

Discrete approximation of a stable self-similar stationary increments process

C. Dombry and N. Guillotin-Plantard

Full-text: Open access


The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known about the context in which such processes can arise. To our knowledge, discretization and convergence theorems are available only in the case of stable Lévy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema first introduced by Cohen and Samorodnitsky, which we consider in a more general setting. Strong relationships with Kesten and Spitzer’s random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process.

Article information

Bernoulli, Volume 15, Number 1 (2009), 195-222.

First available in Project Euclid: 3 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

random scenery random walk self-similarity stable process


Dombry, C.; Guillotin-Plantard, N. Discrete approximation of a stable self-similar stationary increments process. Bernoulli 15 (2009), no. 1, 195--222. doi:10.3150/08-BEJ147.

Export citation


  • [1] Billingsley, P. (1968)., Convergence of Probability Measures. New York: Wiley.
  • [2] Cohen, S. and Samorodnitsky, G. (2006). Random rewards, fractional Brownian local times and stable self-similar processes., Ann. Appl. Probab. 16 1432–1461.
  • [3] Ehm, W. (1981). Sample function properties of multi-parameter stable processes., Z. Wahrsch. Verw. Gebiete 56 195–228.
  • [4] Feller, W. (1971)., Introduction to Probability Theory and Its Applications II. New York: Wiley.
  • [5] Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes., Wahrch. Verw. Gebiete 50 5–25.
  • [6] Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks., Ann. Probab. 19 650–705.
  • [7] Lewis, T.M. and Khoshnevisan, D. (1998). A law of the iterated logarithm for stable processes in random scenery., Stochastic Process. Appl. 74 89–121.
  • [8] Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises ans applications., SIAM Rev. 10 422–437.
  • [9] Mikosch, T., Resnick, S., Rootzen, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab. 12 23–68.
  • [10] Protter, P. (2004)., Stochastic Integration and Differential Equations, 2nd ed. Berlin: Springer.
  • [11] Samorodnitsky, G. and Taqqu, M. (1994)., Stable Non-Gaussian Random Processes. New York: Chapman and Hall.