Suppose $U$ and $V$ are independent and identically distributed (i.i.d), unit-mean exponential random variables. Their product $W = UV$ has finite mean $\mathsf{E}[W] = \mathsf{E}[U]\,\mathsf{E}[V] = 1$ and finite variance $\mathsf{var}(W) = \mathsf{var}(U)\,\mathsf{var}(V) = 1$.
Now consider a sequence of i.i.d. variables $W_i$ (distributed as $W$). By the Central-Limit Theorem, its empirical mean $\bar{W}_n = \sum_{i=1}^{n} W_i$ is such that $\sqrt{n}(\bar{W}_n-n)$ converges in distribution to a normal Gaussian $\mathcal{N}_\mathbb{R}(0,1)$. Therefore we infer that the tail probability $\mathsf{Pr}\{\bar{W}_n \geq t\}$ (for $t > \mathsf{E}[W] = 1$) vanishes as $n\to\infty$ (I suppose Chebyshev's inequality would have sufficed too). From my initial readings into Large Deviations Theory, I would have guessed that it does so exponentially fast.
Now the problem with applying Cramér's Theorem is that the moment-generating function $M(\lambda) = \mathsf{E}[e^{\lambda W}]$ does not seem to exist for $\lambda > 0$. If it existed, my understanding is that I could apply Cramér's Theorem to show that
$$\lim_{n\to\infty}\left\{-\frac{\ln\mathsf{Pr}\{\bar{W}_n \geq t\}}{n} \right\} = \sup_{\lambda \geq 0} \left\{ \lambda t - \ln M(\lambda) \right\}$$
Can I directly conclude from here that $\mathsf{Pr}\{\bar{W}_n \geq t\}$ tends to zero subexponentially as $n\to\infty$? If so, what is the speed of decay (first order term)? How can I determine that? Are there any helpful tools beyond Cramér's Theorem?