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,

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

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

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

Equivalent properties

The following properties are equivalent:

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

See also

• Platykurtic distribution

References

• Kahane, J.P. (1960). "Propriétés locales des fonctions à séries de Fourier aléatoires". Stud. Math. 19. pp. 1–25. .
• Buldygin, V.V.; Kozachenko, Yu.V. (1980). "Sub-Gaussian random variables". Ukrainian Math. J. 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). Adv. Math. 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 Unpublished.