Exercise 2.3.3. Let \(X_{N,i},\ \ 1\le i\ \le N\) be independent random Bernoulli variables with mean \(p_{N,i}\) satisfying \(\mathbf{E}S_N=\sum_{i=1}^Np_{N,i}\rightarrow\lambda\) as \(N\rightarrow\infty\).  By the Poisson limit theorem, Theorem 1.3.4, p. 10, \(S_N\ \rightarrow\ \mathrm{Pois}\left(\lambda\right)\) as \(N\rightarrow\infty\) in distribution.