I have read about Proposition 2.7.1 (Sub-exponential properties) in High Dimensional Probability (HDP), it lists equivalent properties of sub-exponential random variable
(a) The tails of $X$ satisfy $$ \mathbb{P}\{|X| \geq t\} \leq 2 \exp \left(-t / K_1\right) \quad \text { for all } t \geq 0 . $$ (b) The moments of $X$ satisfy $$ \|X\|_{L^p}=\left(\mathbb{E}|X|^p\right)^{1 / p} \leq K_2 p \quad \text { for all } p \geq 1 . $$ (c) The $M G F$ of $|X|$ satisfies $$ \mathbb{E} \exp (\lambda|X|) \leq \exp \left(K_3 \lambda\right) \quad \text { for all } \lambda \text { such that } 0 \leq \lambda \leq \frac{1}{K_3} . $$ (d) The $M G F$ of $|X|$ is bounded at some point, namely $$ \mathbb{E} \exp \left(|X| / K_4\right) \leq 2 . $$
Moreover, if $\mathbb{E} X=0$ then properties a-d are also equivalent to the following one. (e) The MGF of $X$ satisfies $$ \mathbb{E} \exp (\lambda X) \leq \exp \left(K_5^2 \lambda^2\right) \quad \text { for all } \lambda \text { such that }|\lambda| \leq \frac{1}{K_5} $$
Meanwhile, in my other textbook, the definition of sub-exponential is - a random variable $X$ is a sub-Exponential random variable with parameter $(v, \alpha) \in \mathbb{R}^{+} \times \mathbb{R}^{+}$if $$ \forall|\lambda|<1 / \alpha, \quad \mathbb{E}[\exp (\lambda X)] \leq \exp \left(\frac{\lambda^2}{2} v^2\right) $$
Notice that it implies that $v$ and $\alpha$ may be different values. However, when I look at the property 5) in HDP book, i.e.
The $M G F$ of $X$ satisfies $\mathbb{E} \exp (\lambda X) \leq \exp \left(K_5^2 \lambda^2\right) \quad$ for all $\lambda$ such that $|\lambda| \leq \frac{1}{K_5}$
It implies that $v$ = $K_5$ = $\alpha$. I'm confused if it is true that $v$ and $\alpha$ are the same value.