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^{tS_nS_m}) coverge in probablity to E(e^{tXY}) when n,m tends to infinity, where t is real and X and Y are noramlly distrbuted variables with parameters N(np,\sqrt(npq)) and N(mp,\sqrt(npq)), resepectively.

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

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