Sub-Gaussian random variable

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. 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". Adv. Math. 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 Unpublished.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.

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