# Deriving quantiles for $L_1$ using $\textbf{P}\{ L_n > n \cdot a \}$ as n tends to infinity

Given a random variable $$L_n = \sum_{i=1}^n X_i Z_i$$ where $$X_i \sim Bernoulli(p)$$ and $$Z_i \sim \exp(\theta)$$. I assume $$\{Z_i \}$$ is i.i.d. and the sequences $$\{X_i \}$$ and $$\{Z_i \}$$ are independent.
According to Cramér’s large deviation theorem, $$\lim_{n \rightarrow \infty} \frac{1}{n} \log \left( \textbf{P}\{ L_n > n \cdot a \}\right) = - \Lambda^*(a) \\ \Lambda^*(a) = \sup_{\xi \in \mathbb{R}} \left( \xi a - \Lambda (\xi) \right) \\ \Lambda (\xi) = \log \left(\kappa(\xi) \right)$$ where $$\kappa$$ denotes the moment generating function for $$X_1 Z_1$$ and $$a>0$$. Note that $$\kappa(\xi) = \frac{p \theta}{\theta - \xi}+1-p$$. I have calculated the $$\xi^*$$ that solves $$\Lambda^*(a)$$ $$\xi(a,p,\theta)^*= \frac{\theta (2-p) - \sqrt{p \cdot (\theta^2p + \frac{4(p+\theta)}{a}) } }{2(1-p)}.$$ Now, by using the approximation $$\textbf{P}\{ L_n > n \cdot a \} \approx e^{-n \Lambda^*(a)}$$, I want to calculate the $$\alpha$$-quantile $$(VaR_{\alpha})$$, where I think of $$n$$ as "large" and $$\alpha$$, $$\theta$$ and $$p$$ as given. Formally, I can get this far: \begin{align*} \text{VaR}_\alpha(L) = & \text{inf} \{ x \in \mathbb{R} : P(L>x) \leq 1-\alpha \} \\ = & \text{inf} \{ a \in \mathbb{R} : P \left( L>n \cdot a \right ) \leq 1-\alpha ) \\ \approx& \inf \{ a \in \mathbb{R} : e^{-n \Lambda^*(a ) } = 1-\alpha \} \end{align*} where the equality comes from $$\Lambda^*(a)$$ being smooth. Thus I have to find the smallest $$a$$ that solves $$e^{-n\Lambda^*(a )} = \alpha$$ for a "large" n. \begin{align*} e^{-n\Lambda^*(a )} = 1- \alpha \\ e^{-n(\xi^* a - \log (\kappa (\xi^*)) )} =1- \alpha \\ \kappa(\xi^*)^n e^{-n \xi^* a} = 1- \alpha \end{align*} However, I cannot directly solve this equation since $$a$$ is in $$\xi^*$$.
How do I proceed?