Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$
Is there a necessary condition on the $\Omega_i$'s such that under the limit $N \rightarrow \infty$, $E \rightarrow \infty$, while $E/N = \epsilon$ remains finite, one has
$$\lim \frac{\partial^k}{\partial E^k}\log \Omega(N,E) = 0$$ $\forall k\ge 2$?