About the definition of 'sub-gaussian'

Question

I have a question about the definition of subgaussianity.

I have one version of the definition of sub-gaussianity on my textbook here, that is:

Suppose random variable X follows the inequality $\mathbb{E}[\exp(\lambda X)] \leq \exp(\frac{\lambda^2 \sigma^2}{2})$ for $\forall \lambda \in \mathbb{R}$, then we say that $X$ is $\sigma$-subgaussian.

However, this definition automatically results in the $X$ being centered. I also checked the wikipedia on the definition of sub-gaussianity, which seems to have slightly more relaxed condition called the Laplace condition: $$\exists B, b>0, \quad \forall \lambda \in \mathbb{R}, \quad \mathbb{E}[ e^{\lambda(X-\mathrm{E}[X])}] \leq B e^{\lambda^{2} b}$$

My question is: what's the relationship between these two condition? If we have the Laplace condition hold, what can we say about the sub-gaussian parameter of the random variable $X$? Is it automatically $\sqrt{2b}$-subgaussian?

Thanks for any help.

Answer 1

Vershynin's book is a good reference for various equivalent definitions of sub-Gaussianity (and the proofs of their equivalence and keeping track of constants throughout).

Regarding the centering, there isn't an established convention.

• Vershynin only states the $E[e^{\lambda X}] \le e^{\lambda^2 \sigma^2/2}$ definition in the case $E[X]=0$, but also provides other definitions of sub-Gaussianity that don't assume $E[X]=0$.
• Rigollet makes $E[X]=0$ part of the definition of sub-Gaussianity.
• Wikipedia and other texts (like Wainwright) define sub-Gaussianity by applying the MGF condition to the centered random variable $X-E[X]$ instead of $X$.

I'm not sure what textbook you are using, but the MGF condition $E[e^{\lambda X}] \le e^{\lambda^2 \sigma^2 /2}$ is the definition only when $E[X]=0$.

Also, the usage of "$\sigma$-subGaussian" (sometimes elsewhere as "$\sigma^2$-subGaussian" like in Rigollet's text) isn't standardized, and may differ from context to context.