Bounded Stationary Stable Processes and Entropy

  • John P. Nolan
Part of the Progress in Probabilty book series (PRPR, volume 25)


In this paper we will show that any stationary or stationary increment p-stable process, 1≤p<2, that is sample bounded has a finite metric entropy integral. The result is an application of Talagrand’s work on majorizing measures for stable processes [7]. We combine this result with earlier results to give necessary conditions for a stationary increment stable process to have a.s. bounded or a.s. continuous sample paths.


Sample Path Stable Process Stationary Increment Finite Dimensional Distribution Symmetric Stable Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Cambanis, S., Nolan, J. P. and Rosinski, J. “On the oscillation of infinitely divisible and some other processes.” To appear in Stochastic Processes and Their Applications (1990).Google Scholar
  2. [2]
    Cambanis, S. and Soltani, A. R. “Prediction of stable processes: spectral and moving average representations.” Z. Wahrsch. verw. Gebiete 66, 593–612 (1983).MathSciNetCrossRefGoogle Scholar
  3. [3]
    Hardin, C. D. “On the spectral representation of symmetric stable processes.” J. Multivar. Analysis 12,385–401 (1982).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Marcus, M. B. and Pisier, G. “Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes.” Acta Math. 152, 245–301 (1984).MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Marcus, M. B. and Pisier, G. Random Fourier Series with Applications to Harmonic Analysis. Princeton University Press, Princeton, NJ (1981).MATHGoogle Scholar
  6. [6]
    Nolan, J. P. “Continuity of symmetric stable processes.” J. Multivar. Analysis 29, 84–93 (1989).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Talagrand, M. “Necessary conditions for sample boundedness of p-stable processes.” Annals of Probability 16, 1584–1595 (1988).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • John P. Nolan
    • 1
  1. 1.Department of Mathematics and StatisticsAmerican UniversityUSA

Personalised recommendations