the limiting distribution of a sum of random variables I'm thinking about the following old exam problem:
Given a triangular array of random variables $\{X_{nk}\}$, $1\leq k\leq r_n$, $n\in\mathbb{N}$, with $p_{nk}=P(X_{nk}=1)$ and $P(X_{nk}=0)=1-p_{nk}$. If 
$$
\sum_{k=1}^{r_n}p_{nk}\rightarrow\lambda, max_{k\leq r_n} p_{nk}\rightarrow0,
$$
what is the limiting distribution of $\sum_{k=1}^{r_n}X_{nk}$?
I tried to use the theorem that $\frac{S_n-E(S_n)}{b_n}\rightarrow0$ in probability if $\frac{Var(S_n)}{b_n^2}\rightarrow0$ in the first place, but it doesn't seem to be working since it will need to add a denominator which goes to infinity and the distribution of $S_n$ is still unknown. I also tried Lindeberg-Feller theorem, and it has a similar problem as above. For these both thoughts, I don't know how to use the condition that maximum of $p_{nk}$ goes to zero. Could I ask for a hint? Thanks for any help.
 A: These are exactly the conditions for the Lindberg Feller theorem for Poisson convergence. The intuition is that you have a sequence of very rare events whose total expectation is not small. See for example section 4 of this link. The full theorem has an extra condition:
$$\sum_{m=1}^n\epsilon_{n,m}\rightarrow 0,$$
where $\epsilon_{n,m}=P(X_{n,m}\geq 2)$. In your case it's trivially satisfied since your variables are Bernoulli.
A: Hint: Poisson distribution and the law of small numbers. Consider the simpler case $X_{nk} \sim Ber(1/n)$ that is $P(X_{nk} = 1) = 1/n$, $P(X_{nk} = 0) = 1 - \frac{1}{n}$.
Assume independence of the random variables. Let $r_n = n$.  $Y_n = \sum_{k=1}^n X_{nk}$. Let $Z \sim \text{Poi}(1)$
Calculate: $$\Bbb{P}(Y_n = k) = {n\choose k} \bigg(\frac{1}{n}\bigg)^k \bigg(1 - \frac{1}{n}\bigg)^{n-k} \xrightarrow[n \to \infty]{} \frac{e^{-1}}{k!} = \Bbb{P}(Z = 1)$$
From here you can see that $Y_n\to Z$ in distribution.
remark Lindberg feller will not work in this context since you do not have small jumps, that is the random variables are not symbiotically negligible. 
