# Sub-Gaussian and "nearly" sub-Gaussian random variables

Define $$\Psi_{X}(\lambda) = \log E e^{\lambda X}$$, and suppose that $$EX=0$$. We say that the random variable $$X$$ is sub-Gaussian with variance factor $$v$$ if: $$\Psi_{X}(\lambda) \leq v\lambda^{2}/2$$ for all $$\lambda \in \mathbb{R}$$. In other words, the moment generating function of $$X$$ is "dominated" by the moment generating function of a mean zero Gaussian random variable with variance $$v$$.

As far as I know, this indicates that sub-gaussian random variables with variance factor $$v$$ have tails that are "thinner" than the tails of a Gaussian random variable with variance $$v$$.

Sub-Gaussian random variables are particularly important in the study of concentration inequalities.

Now I have found cases (for example, equation (1.1) in the supplementary material of the paper Spokoiny (2012)) where concentration inequalities are stated for the case when: $$\Psi_{X}(\lambda) \leq v\lambda^{2}/2$$ for all $$\lambda$$ satisfying $$|\lambda|\leq c$$ for some $$c$$.

I am wondering if anyone can offer an interpretation of this? In particular, what can we say about the tails of a random variable that satisfy the sub-Gaussian inequality only for values of $$\lambda$$ satisfying $$|\lambda|\leq c$$?

A random variable is $$K$$-sub-exponential if $$P(|X| \geq t) \leq 2 \exp(-K/t)$$. One of its equivalent definition is that $$\Psi_X(t) \leq C^2 \lambda^2$$ for all $$|\lambda| \leq \frac{1}{C}$$. See Proposition 2.7.1 in HDP. You can consult Section 2.7 titled 'Sub-Exponential Distributions' in the HDP to know more.