Suppose that $S_n$ and $S_m$ are two random binomial variables, which are independent and with the same distribution parameter $p$. I am wondering, in which cases $E(e^{t S_n S_m})$ converges to $E(e^{tXY})$ when $n, m$ tend to infinity, where $t$ is real and $X$ and $Y$ are normally distributed variables with parameters $N(np,\sqrt{npq})$ and $N(mp,\sqrt{mpq})$, respectively.

However, it is clear that $e^{tS_nS_m}$ converges in probability to $e^{tXY}$, according to the Moivre-Laplace theorem, but I am not sure that any of the sufficient conditions which can be found in literature are satisfied considering the expectation.

In fact, I am trying to prove $E(e^{tS_nS_m}) \sim E(e^{tXY})$ when $n, m$ tends to infinity.