Let $(X_n)$ be i.i.d double exponential random variables, with PDF $\frac1{2a}e^{-|x|/a}$, thus $E(X_n) = 0$, $E(|X_n|) = a$ and $V(X_n)=2a^2$. Consider the sample means $$\bar{X}_n=\frac1n\sum\limits_{i=1}^nX_i\qquad\bar{Y}_n=\frac1n\sum\limits_{i=1}^n|X_i|.$$ One is interested in the asymptotic distribution of $$R_n=\frac{\bar{Y}_n - a}{\bar{X}_n}.$$
I want to use the delta method, but the expectation of $\bar{X}_n$ is $0$, which causes the problem when dividing.
Any other idea?