Reference request: Poisson approximation to the binomial distribution in total variation

Question

It is known and not hard to prove (see e.g. this answer) that the binomial distribution with parameters $n,p$ converges in total variation to the Poisson distribution with parameter $\lambda\in(0,\infty)$ whenever $n\to\infty$ and $np\to\lambda$. However, I have not been able to find this fact in a completely ready-to-use form in the published literature. I would appreciate a reference to such a source.

Answer 1

It should be in any Introduction to Probability book.

For example, a quick search turned up a book by Michael J. Evans and Jeffrey S. Rosenthal called Probability and Statistics and this is Example 2.3.6

Reaching out my arm to something more widely known, Kendall and Stuart's The Advanced Theory of Statistics (2nd edition), it is in section 5.8 in Volume 1.

Answer 2

This can be found (in a maybe overkill form, as the paper is concerned with tight bounds, with the "right" constants) in a paper of Roos:

Roos, Bero. "Sharp constants in the Poisson approximation." Statistics & probability letters 52, no. 2 (2001): 155-168. https://doi.org/10.1016/S0167-7152(00)00208-X

(see specifically Theorem 2). As mentioned, this is a bit overkill, as it deals with more general bounds on the distance between Poisson Binomial and Poisson (not just Binomial and Poisson).

Note that you can combine this with bounds on the TV between Poisson distributions (parameters $\lambda$ and $np$ in your case) to get what you want what the triangle inequality. For that second part, see for instance Section 2.1 (eq (2.2)) of

Adell, José A., and Pedro Jodrá. "Exact Kolmogorov and total variation distances between some familiar discrete distributions." Journal of Inequalities and Applications 2006, no. 1 (2006): 64307. https://doi.org/10.1155/JIA/2006/64307

Answer 3

Le Cam’s Theorem takes care of this, clearly it is overkill but you seem OK with that. See the Wikipedia page here and the references therein.