I've come across the following question. Say we have two families of random variables, $X_N$ and $Y_N$, such that $\mathbb{E} X_N=\mathbb{E} Y_N=0$ and $\mathbb{E}X_N^2=1$. Now assume that for $\lambda\in \mathbb{C}$: $$\mathbb{E} e^{\lambda X_N}\mathbb{E} e^{\lambda Y_N}=e^{a_N^2\lambda^2/2}(1+o(1))$$ where $o(1)$ is uniform over compact subsets of $\mathbb{C}$, and of course $a_N^2=\mathbb{E} Y_N^2+1$.
My question is whether from this you can conclude that $X_N\Rightarrow \cal{N}(0,1)$?
If $a_N^2$ is bounded this follows from Cramer's Decomposition Theorem, but I cannon't either prove or disprove the statement in the case where $a_N^2$ is unbounded.
Any help would be appreciated.