Sub-Gaussian distribution

In probability theory, a sub-Gaussian distribution is a probability distribution with strong tail decay. Informally, the tails of a sub-Gaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian.

Formally, the probability distribution of a random variable X is called sub-Gaussian if there are positive constants C, v such that for every t > 0,

The sub-Gaussian random variables with the following norm form a Birnbaum–Orlicz space:

Equivalent definitions

The following properties are equivalent:

  • The distribution of X is sub-Gaussian
  • -condition: for some a > 0, .
  • Laplace transform condition: for some B, b > 0, holds for all .
  • Moment condition: for some K > 0, for all p > 1.
  • Union bound condition: for some c > 0, for all n > c, where are i.i.d copies of X.

See also

References
  • Kahane, J.P. (1960). "Propriétés locales des fonctions à séries de Fourier aléatoires". Studia Mathematica. Vol. 19. pp. 1–25. .
  • Buldygin, V.V.; Kozachenko, Yu.V. (1980). "Sub-Gaussian random variables". Ukrainian Math. J. Vol. 32. pp. 483–489. .
  • Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach Spaces. Springer-Verlag.
  • Stromberg, K.R. (1994). Probability for Analysts. Chapman & Hall/CRC.
  • Litvak, A.E.; Pajor, A.; Rudelson, M.; Tomczak-Jaegermann, N. (2005). "Smallest singular value of random matrices and geometry of random polytopes" (PDF). Advances in Mathematics. Vol. 195. pp. 491–523.
  • Rudelson, Mark; Vershynin, Roman (2010). "Non-asymptotic theory of random matrices: extreme singular values". arXiv:1003.2990.
  • Rivasplata, O. (2012). "Subgaussian random variables: An expository note" (PDF). Unpublished.
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