Sub-Gaussian distribution

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

Formally, $X$ is called sub-Gaussian if there are positive constants $C,v$ such that for any $t>0$ :

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

The sub-Gaussian random variables with the following norm:

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

form a Birnbaum–Orlicz space.

Equivalent properties

The following properties are equivalent:

• $X$ is sub-Gaussian
• $\psi _{2}$-condition: $\exists a>0\ Ee^{aX^{2}}<+\infty$.
• Laplace transform condition: $\exists B,b>0\ \forall \lambda \in \mathbb {R} \ \ Ee^{\lambda X}\leq Be^{\lambda ^{2}b}$.
• Moment condition: $\exists K>0\ \forall p\geq 1\ \left(E|X|^{p}\right)^{1/p}\leq K{\sqrt {p}}$ .

References

• Kahane, J.P. (1960). "Propriétés locales des fonctions à séries de Fourier aléatoires". Stud. Math. Vol. 19. pp. 1–25. [1].
• Buldygin, V.V.; Kozachenko, Yu.V. (1980). "Sub-Gaussian random variables". Ukrainian Math. J. Vol. 32. pp. 483–489. [2].
• 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". Adv. Math. Vol. 195. pp. 491–523. PDF.
• Rudelson, Mark; Vershynin, Roman (2010). "Non-asymptotic theory of random matrices: extreme singular values". arXiv:1003.2990. PDF.
• Rivasplata, O. (2012). "Subgaussian random variables: An expository note". Unpublished. PDF.