Let $X_1, X_2, . . .$ be independent, $L(a)$-distributed random variables, and let $N \in Po(m)$ be independent of $X_1, X_2, . . . . $
Determine the limit distribution of $S_N = X_1 + X_2 + · · · + X_N$ (where $S_0 = 0)$ as $m \to \infty$ and $a \to 0$ in such a way that $m · a^2 \to 1$.
I think that using the moment generating might be the key. I did try to look for $\lim_{n \to \infty}M_{S_N}(t)$ and condition on N being a Poisson.
$$\lim_{n \to \infty}M_{S_N}(t)=\lim_{n \to \infty}E(e^{tS_N})=\lim_{n \to \infty}EE(e^{tS_N}|N)= \lim_{n \to \infty}P(N=n)E(e^{tS_n})=\lim_{n \to \infty} \frac{m^ne^{-m}}{n!}(\frac{a^2}{a^2-t^2})^n=\lim_{n \to \infty} \frac{(ma^2)^ne^{-m}}{n!(a^2-t^2)^n}=e^{-m}/n!(a^2-t^2)^n $$
Any hint would be appreciated.