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,

$\operatorname {P} (|X|>t)\leq Ce^{-vt^{2}}.$

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

$\|X\|_{\psi _{2}}=\inf\{s>0\mid \operatorname {E} e^{(X/s)^{2}}-1\leq 1\}.$

Equivalent definitions

The following properties are equivalent:

• The distribution of X is sub-Gaussian
• $\psi _{2}$-condition: for some a > 0, $\operatorname {E} e^{aX^{2}}<+\infty$.
• Laplace transform condition: for some B, b > 0, $\operatorname {E} e^{\lambda (X-\operatorname {E} [X])}\leq Be^{\lambda ^{2}b}$ holds for all $\lambda$.
• Moment condition: for some K > 0, $\operatorname {E} |X|^{p}\leq K^{p}p^{p/2}$ for all p > 1.
• Union bound condition: for some c > 0, $\ \operatorname {E} [\max\{|X_{1}-\operatorname {E} [X]|,\ldots ,|X_{n}-\operatorname {E} [X]|\}]\leq c{\sqrt {\log n}}$ for all n > c, where $X_{1},\ldots ,X_{n}$ are i.i.d copies of X.

See also

• Platykurtic distribution

References

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• Buldygin, V.V.; Kozachenko, Yu.V. (1980). "Sub-Gaussian random variables". Ukrainian Mathematical Journal. Vol. 32. pp. 483–489. doi:10.1007/BF01087176.
• 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. doi:10.1016/j.aim.2004.08.004.
• Rudelson, Mark; Vershynin, Roman (2010). "Non-asymptotic theory of random matrices: extreme singular values". Proceedings of the International Congress of Mathematicians 2010. pp. 1576–1602. arXiv:1003.2990. doi:10.1142/9789814324359_0111.
• Rivasplata, O. (2012). "Subgaussian random variables: An expository note" (PDF). Unpublished.