Sub-Gaussian random variable

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 constants $C, v$ such that for any $t>0$ :

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

The sub-Gaussian random variables with the following norm:

$\|X\|_{\psi_2} = \inf\{s>0\mid E e^{(X/s)^2} -1 \leq 1\}.$

form a Birnbaum–Orlicz space.

Equivalent properties

The following properties are equivalent:

• $X$ is sub-Gaussian
• $\psi_2$-condition: $\exists a>0 \ E e^{aX^2}<+\infty$.
• Laplace transform condition: $\exists B, b>0 \ \forall \lambda \in \mathbb{R} \ \ E e^{\lambda X} \leq Be^{\lambda^2 b}$.
• Moment condition: $\exists K>0 \ \forall p \geq 1 \ \left ( E |X|^p \right )^{1/p} \leq K \sqrt{p}$ .

References

• Rudelson, Mark; Vershynin, Roman (2010). "Non-asymptotic theory of random matrices: extreme singular values". arXiv:1003.2990. PDF.

• Ledoux, Michel; Talagrand, Michel (2013). Probability in Banach Spaces: isoperimetry and processes. Springer Science & Business Media.