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 $|\lambda|\leq c$?
It means that the random variable is sub-exponential, i.e., has heavier tails than the sub-Gaussian distribution.
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[1]. You can consult Section 2.7 titled 'Sub-Exponential Distributions' in the HDP[1] to know more.
[1]: High-Dimensional Probability, Roman Vershynin. https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html