Poisson Convergence Example

For any $n\ge1,$ let $X_{n1},...,X_{nn}$ be independent with $$P\{X_{nk}=5\}=1-P\{X_{nk}=0\}=p_{nk}, \ k=1,...,n$$.

Assume that $$\lim_{n\rightarrow \infty} \max_{1\le k \le n} p_{nk}=0$$ and $$\lim_{n\rightarrow \infty} \sum_{k=1}^{n} p_{nk} =2$$.

Find the limit in distribution of $S_n=X_{n1}+\cdots+X_{nn}$.

I know this converges to a poisson distribution. This would easy to show using the characteristic function if the random variables were identically distributed but they are not in this case.

You can prove such assertions using rare event laws. See for example Theorem 7.13 of this reference. The proof follows through using characteristic functions, since your random variables are essentially Bernoulli and independent after rescaling $Y_i:=X_i/5$, and will have characteristic function $1+p_{n,m}(e^{it}-1)$. You will use the maximum assertion and the sum convergence assertion when expanding the product of characteristic functions, to bound higher order terms.