Encyclopedia of Database Systems

2018 Edition
| Editors: Ling Liu, M. Tamer Özsu

Stable Distribution

Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_367


Lévy skew α-stable distribution


A random variable Z is said to follow a symmetric α-stable distribution [ 13, 15], where 0 < α ≤ 2, if the Fourier transform of its probability density function f Z ( z) satisfies
$$ {\int}_{-\infty}^{\infty }{e}^{\sqrt{-1} zt}{f}_Z(z) dt={e}^{-d\left|t\right|{}^{\alpha }},\,\, 0<\alpha \le 2 $$
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Achlioptas D. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J Comput Syst Sci. 2003;66(4):671–87.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon N, Matias Y, Szegedy M. The space complexity of approximating the frequency moments. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing; 1996. p. 20–9.Google Scholar
  3. 3.
    Cormode G, Datar M, Indyk P, Muthukrishnan S. Comparing data streams using hamming norms (how to zero in). In: Proceedings of the 28th International Conference on Very Large Data Bases; 2002. p. 335–45.CrossRefGoogle Scholar
  4. 4.
    Datar M, Immorlica N, Indyk P, Mirrokn VS. Locality-sensitive hashing scheme based on p-stable distributions. In: Proceedings of the 20th Annual Symposium on Computational Geometry; 2004. p. 253–62.Google Scholar
  5. 5.
    Donoho DL. Compressed sensing. IEEE Trans Inform Theory. 2006;52(4):1289–306.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fama EF, Roll R. Parameter estimates for symmetric stable distributions. J Am Stat Assoc. 1971;66(334):331–8.zbMATHCrossRefGoogle Scholar
  7. 7.
    Indyk P. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J ACM. 2006;53(3):307–23.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Indyk P, Motwani R. Approximate nearest neighbors: towards removing the curse of dimensionality. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing; 1998. p. 604–13.Google Scholar
  9. 9.
    Johnson WB, Lindenstrauss J. Extensions of Lipschitz mapping into Hilbert space. Contemp Math. 1984;26(189–206):1–1.1.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Li P. Very sparse stable random projections for dimension reduction in lα (0 < α ≤ 2) norm. In: Proceedings of the 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining; 2007.Google Scholar
  11. 11.
    Li P. Estimators and tail bounds for dimension reduction in lα (0 < α ≤ 2) using stable random projections. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms; 2008.Google Scholar
  12. 12.
    Muthukrishnan S. Data streams: algorithms and applications. Found Trends Theor Comput Sci. 2005;1(2):117–236.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Samorodnitsky G, Taqqu MS. Stable Non-Gaussian random processes: Chapman & Hall; 1994.Google Scholar
  14. 14.
    Vempala S. The random projection method. Providence: American Mathematical Society; 2004.zbMATHGoogle Scholar
  15. 15.
    Zolotarev VM. One-dimensional stable distributions. Providence: American Mathematical Society; 1986.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Cornell UniversityIthacaUSA

Section editors and affiliations

  • Divesh Srivastava
    • 1
  1. 1.AT&T Labs - ResearchAT&TBedminsterUSA