Sub-Gaussian distribution

In probability theory, a sub-Gaussian random variable, is a random variable with strong tail decay property. 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

• Template:Cite article
• Template:Cite article [1].
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach Spaces: isoperimetry and processes. Springer-Verlag.
• 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: 491–523 PDF.
• Rudelson, Mark; Vershynin, Roman (2010). "Non-asymptotic theory of random matrices: extreme singular values". arXiv:1003.2990. PDF.