# About the definition of 'sub-gaussian'

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.

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.

• Thanks for your detailed answer and reference. I have looked at the book you suggest but I couldn't find any result addressing the constant $B$ before the exponential term. Once we derive that $\mathbb{E}\left[e^{\lambda(X-\mathrm{E}[X])}\right] \leq B e^{\lambda^{2} b}$, can we simply say that $X$ is $\sqrt{2b}$-subgaussian and ignore the disturbing $B$ constant before? Commented Aug 28, 2021 at 8:27
• @JamieCarr Oh I see your specific concern now. If you walk through the proof of "(v) $\implies$ (i)" in Vershynin's text with an extra factor of $B$, you will find that you still get the concentration in (i) but with a constant different than $2$ in the front of the bound. Note that Wikipedia's probability bound has a general $C$ instead of specifically $2$. So there is a subtle difference in the definitions, but I think in practice most would not care too much about this constant factor; it is the exponential decay that matters. Commented Aug 28, 2021 at 16:54
• @JamieCarr Also note that Wikipedia only gives a definition for "sub-Gaussian" without any sort of parameterization by $\sigma$ or $\sigma^2$, so even disregarding the difference in my previous comment, you cannot use it to say something like "$\sqrt{2b}$-subGaussian." Commented Aug 28, 2021 at 16:55