Synonyms
Lévy skew α-stable distribution
Definition
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 fZ (z) satisfies
where d > 0 is the scale parameter. This is denoted by Z ∼ S(α, d).
There is an equivalent definition. A random variable Z follows a symmetric α-stable distribution if, for any real numbers, C1 and C2,
where Z1 and Z2 are independent copies of Z, and the symbol “\( \overset{d}{=} \)” denotes equality in distribution.
The probability density function fZ (z) can be obtained by taking inverse Fourier transform of 1. In particular, fZ (z) can be expressed in closed-forms when α = 2 (i.e., the normal distribution) and α= 1 (i.e., the...
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Achlioptas D. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J Comput Syst Sci. 2003;66(4):671–87.
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.
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.
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.
Donoho DL. Compressed sensing. IEEE Trans Inform Theory. 2006;52(4):1289–306.
Fama EF, Roll R. Parameter estimates for symmetric stable distributions. J Am Stat Assoc. 1971;66(334):331–8.
Indyk P. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J ACM. 2006;53(3):307–23.
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.
Johnson WB, Lindenstrauss J. Extensions of Lipschitz mapping into Hilbert space. Contemp Math. 1984;26(189–206):1–1.1.
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.
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.
Muthukrishnan S. Data streams: algorithms and applications. Found Trends Theor Comput Sci. 2005;1(2):117–236.
Samorodnitsky G, Taqqu MS. Stable Non-Gaussian random processes: Chapman & Hall; 1994.
Vempala S. The random projection method. Providence: American Mathematical Society; 2004.
Zolotarev VM. One-dimensional stable distributions. Providence: American Mathematical Society; 1986.
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Li, P. (2018). Stable Distribution. In: Liu, L., Özsu, M.T. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8265-9_367
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