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

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.

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

• Thank you for this answer. I don't have the Roos paper right now, but its description on MathScinet contains this: "The distribution of the sum of independent Bernoulli random variables is approximated by the Poisson law with the same mean." On the other hand, I have $np\to\lambda$, rather than $np=\lambda$. Are you sure the Roos paper contains what asked about? – Iosif Pinelis Feb 23 at 3:01
• @IosifPinelis You are right, in that the paper proves bounds (depending on $p$, and, more generally, on the ratio $\frac{\sum_{j=1}^n p_j^2}{\sum_{j=1}^n p_j}$) between Binomial (more generally, Poisson binomial) and Poisson with the same mean. However, you can use the triangle inequality to easily get what you want, combining it with bounds on the TV distance between poisson distributions with mean $np$ and $\lambda$, respectively. – Clement C. Feb 23 at 4:34
• I'm adding some refs to my answer, accordingly. – Clement C. Feb 23 at 4:35
• Thank you for this additional reference. However, my question is about "a completely ready-to-use form", which would state exactly what is stated in my answer: "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 $np\to\lambda$." A result more general than this would also be OK. But an easy combination of known results is not what I am looking for -- everything here is very easy to prove to begin with, that is not a problem. – Iosif Pinelis Feb 23 at 4:57
• @IosifPinelis That's fair. – Clement C. Feb 23 at 5:36

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.

• Thank you for your answer. I am aware of Le Cam's theorem. However, there the mean of the Poisson distribution is the same as the mean of the approximated distribution. So, unfortunately, Le Cam's theorem does not cover the statement in my question. – Iosif Pinelis Feb 23 at 5:01
• @Iosif Maybe I’m missing the point of the question, but the result you want is so immediate from Le Cam’s theorem that I can’t imagine anyone raising a fuss if you cite it as justification for your fact. More likely is that they would wonder why you would feel the need to cite something at all when the result is common knowledge. – guy Feb 23 at 5:59

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.

• Thank you for this answer. I have looked at Example 2.3.6 in the book by Evans and Rosenthal, but it seems to have nothing about the convergence in total variation, and also only the case $p=\lambda/n$ is considered there. – Iosif Pinelis Feb 23 at 3:04
• The page you indicate in Evans and Rosenthal is only about convergence at any integer $x.$ That's not the same as total variation distance. – Michael Hardy Feb 23 at 5:07