I am working through High Dimensional Statistics (Wainwright) and am stuck at 2.14, trying to show that a random variable $X$ is sub-exponential. The context is as follows:
Definition of Sub-Exponential
A random variable $X$ with mean $\mu = \mathbb{E}[X]$ is defined as sub-exponential if there are non-negative parameters $(\nu, \alpha)$ such that: $$\mathbb{E}[e^{\lambda(X-\mu)}] \leq e^{\frac{\nu^2\lambda^2}{2}} \;\;\;\;\;\; \text{for all } |\lambda| < \frac{1}{\alpha}$$ Example
Let $Z \sim \mathcal{N}(0, 1)$, and consider the random variable $X = Z^2$. For $\lambda < \frac{1}{2}$ we have: $$\mathbb{E}[e^{\lambda(X-1)}] = \frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{\lambda (z^2 - 1)} e^{\frac{-z^2}{2}}dz$$ $$ = \frac{e^{-\lambda}}{\sqrt{1 - 2 \lambda}}$$ Following some calculus, we find that: $$\frac{e^{-\lambda}}{\sqrt{1 - 2 \lambda}} \leq e^{2\lambda^2} = e^{4\lambda^2/2}, \;\;\;\; \text{for all } |\lambda| < \frac{1}{4} \tag{2.14}\label{2.14}$$ Which shows that $X$ is sub-exponential with parameters $(\nu, \alpha) = (2,4)$.
I am struggling to prove the inequality in 2.14. I have been able to derive the the left hand side, and have created some plots to show that it is indeed true, but I don't have the intuition for how the author arrived at it. In other words, I am trying to figure out the intuition, thought process, and steps to take when trying to prove something like this.
I have tried to look at the taylor series expansion of the left hand side, but again don't see how the author made the leap to compare it with $e^{2\lambda^2}$.
The logarithm is $-\lambda-\tfrac12\ln(1-2\lambda)=2\lambda^2+O(\lambda^3)$, with all $o(\lambda^2)$ terms having positive coefficient.