Let $X_n$ be a sequence of random variables with $X_n\sim\mathrm{Bin} \left( n, p(n) \right)$. I'm trying to understand how the limiting behavior of this sequence of distributions depends on the asymptotic behavior of $p(n)$.
It is well known that if $np(n)\rightarrow \lambda$ for a constant $\lambda$ then $X_n \stackrel{d}{\rightarrow} \mathrm{Pois}(\lambda)$. It is also well known that if $p(n)\rightarrow c$ for a constant $0<c<1$ then $\frac{X_n-cn}{\sqrt{c(1-c)n}} \stackrel{d}{\rightarrow} \mathrm{N}(0,1)$. But what happens in the great asymptotic range between those two edge cases? What about $p(n)=\frac{1}{n^{1/6}}$ or $p(n)=\frac{1}{\sqrt{n}}$ or even $p(n)=\frac{\ln n}{n}$, for example?