I'm hoping that somebody can help clarify a question I had regarding the Poisson approximation and its applications. My textbook presents the following theorem, which I'm having trouble making sense of. My assumption is that the first capital $N$ is meant to be a lowercase $n$, and that all subsequent $N$'s represent a discrete random variable with a binomial distribution:
Theorem 2.5. Consider independent events $A_i$, $i=1,2,...,n$, with probabilities $p_i=P(A_i)$. Let $N$ be the number of events that occur, let $\lambda=p_1+···+p_n$, and let $Z$ have a Poisson distribution with parameter $\lambda$. Then, for any set of integers $B$,$$\left\vert P(N\in B)-P(Z\in B)\right\vert\leq\sum_{i=1}^n p_i^2\tag{2.14}$$ We can simplify the right-hand side by noting $$\sum_{i=1}^n p_i^2\leq \max_i p_i \sum_{i=1}^n p_i=\lambda\max_ip_i$$ This says that if all the $p_i$ are small then the distribution of $N$ is close to a Poisson with parameter $\lambda$. Taking $B=\{k\}$, we see that the individual probabilities $P(N=k)$ are close to $P(Z=k)$, but this result says more. The probabilities of events suchas $P(3\leq N\leq 8)$ are close to $P(3\leq N\leq 8)$ and we have an explicit bound on the error.
The text then refers back to an example comparing the exact probability of obtaining exactly one double $6$ in twelve rolls of a pair of dice to the corresponding Poisson approximation:
Suppose we roll two dice $12$ times and we let $D$ be the number of times a double $6$ appears. Here, $n=12$ and $p=1/36$, so $np=1/3$. We now compare $P(D=k)$ with the Poisson approximation for $k=1$. $$k=1 \text{ exact answer:}\,\,\,\,\,\,P(D=1)=\left(1-\frac{1}{36}\right)^{12} =0.7132$$ $$\text{Poisson approximation:}\,\,\,\,\,\,P(D=1)=e^{-1/3}=0.7165$$
For a concrete situation, consider [the example above], where $n=12$ and all the $p_i=1/36$. In this case the error bound is $$\sum_{i=1}^{12} p_i^2 =12\left(\frac{1}{36}\right) ^2=\frac{1}{108}=0.00926$$ while the error for the approximation for $k=1$ is $0.0057$.
From this context, my thought is that the error bound they allude to is specifically that for the Poisson approximation to the binomial distribution, and not the Poisson approximation to some other type of distribution. Can somebody with a more complete understanding of the Poisson distribution (and its relationship to the binomial) confirm or refute this assertion? I'd also be curious to know where the proof of this theorem comes from, as my text doesn't offer any obvious reference.
Citation: Elementary Probability for Applications, Rick Durrett.