Poisson approximation by coupling

Rinaldo B. Schinazi

arxiv: 2605.01894 · v1 · submitted 2026-05-03 · 🧮 math.PR

Poisson approximation by coupling

Rinaldo B. Schinazi This is my paper

Pith reviewed 2026-05-09 16:39 UTC · model grok-4.3

classification 🧮 math.PR

keywords Poisson approximationcouplingbinomial distributionelementary probabilitytotal variation distanceapproximation bounds

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The pith

A coupling between binomial and Poisson random variables bounds their difference for any finite n and p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the usual Poisson approximation to the binomial can be obtained by constructing a joint distribution, or coupling, for the two random variables instead of taking limits as n grows large. This produces a stronger result that holds without asymptotic assumptions and relies only on basic probability facts such as indicator variables and simple inequalities. A reader would care because the method gives an explicit, non-asymptotic way to measure how close the distributions are. The argument avoids convergence theorems entirely and works uniformly over all parameter values.

Core claim

By constructing a coupling of a Bin(n,p) random variable and a Pois(np) random variable on the same probability space, the probability that the two variables differ can be bounded directly; this bound controls the total variation distance between their laws for every finite n and p, using only elementary probability.

What carries the argument

The coupling of binomial and Poisson random variables, which places both on one space so that their difference occurs with controlled probability.

If this is right

The approximation error is controlled for every finite n and p rather than only in the limit.
The proof uses nothing beyond indicator sums and basic probability inequalities.
The same coupling argument extends immediately to other parameter regimes where the usual limit theorem does not apply directly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique may simplify teaching of Poisson approximation by removing the need for limit theorems at an early stage.
Similar couplings could be sought for other classical approximations such as the normal limit for the binomial.
The method supplies a concrete, computable bound rather than an existence statement from convergence in distribution.

Load-bearing premise

A suitable coupling between the binomial and Poisson random variables can be built using only elementary probability without hidden assumptions or advanced tools.

What would settle it

An explicit pair n and p for which every possible coupling yields a difference probability that cannot be bounded by an elementary expression in n and p.

read the original abstract

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$ goes to infinity and $np$ converges to some $\lambda$. In this note we use instead the coupling technique to show a much more general result. Moreover, we only use elementary results from probability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

An elementary coupling argument delivers an explicit non-asymptotic bound between binomial and Poisson, but the advance is modest and mostly pedagogical.

read the letter

The paper's core move is to couple a binomial as a sum of independent Bernoullis with a sum of independent Poissons and bound the probability they differ by a direct calculation using only the inequality 1 - e^{-p} - p e^{-p} ≤ p²/2. This produces a finite-n, finite-p total-variation bound without taking limits or invoking generating functions or Poisson processes. That is the actual content: a self-contained, elementary derivation of a stronger statement than the usual convergence theorem found in first courses. It does this cleanly and stays within the definitions of the two distributions plus independence and finite sums, which is what the abstract promises. The construction is reproducible from the description and does not rely on hidden tools or circular steps. The bound itself is not claimed to be optimal, and the paper does not compare it numerically to Stein-Chen or other known couplings, so the practical improvement over existing elementary proofs is not quantified. Because the result is framed as a note rather than a new framework, the contribution sits at the level of a clean teaching example rather than a reorganization of approximation theory. The citation pattern is light and appropriate for the scope. A reader who wants an explicit, limit-free bound that can be taught in an undergraduate course will find the argument useful and self-contained. A reader looking for sharper constants or new applications will not. The work is coherent on its own terms and shows clear engagement with the standard elementary toolkit, so it deserves a referee's time in a venue that publishes short notes on probability methods. I would bring it to a reading group only if the group is explicitly working on coupling techniques or undergraduate exposition.

Referee Report

2 major / 1 minor

Summary. The manuscript uses a coupling argument to derive an explicit, non-asymptotic bound between the binomial distribution Bin(n,p) and the Poisson distribution Pois(np) that holds for all finite n and p. The construction writes the binomial as a sum of independent Bernoullis, introduces an auxiliary collection of independent Poisson(p) random variables, and controls P(X≠Y) via the elementary inequality 1−e^{-p}−p e^{-p} ≤ p²/2, relying only on definitions, independence, and finite sums.

Significance. If the coupling is correctly constructed and the bound is made explicit, the result strengthens the usual n→∞ convergence by supplying a concrete finite-n,p guarantee obtained without generating functions, Poisson processes, or limit theorems. The elementary character of the argument is a genuine pedagogical and methodological strength.

major comments (2)

The manuscript does not state the final explicit bound on the distance (e.g., total-variation or P(X≠Y)) obtained from the coupling; without this statement the claim of a 'much more general result' cannot be evaluated against standard bounds such as Le Cam's inequality.
The joint construction of the coupling (X binomial, Y Poisson) must be written out with the precise definition of the auxiliary variables and the verification that the marginals are exactly Bin(n,p) and Pois(np); the current description leaves open whether any measurability or independence step requires tools beyond the elementary facts invoked.

minor comments (1)

The abstract refers to 'a much more general result' without indicating whether the generality extends beyond the finite-n,p regime or merely removes the asymptotic restriction; a single clarifying sentence would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments identify opportunities to strengthen the explicitness of the main result and the clarity of the construction. We address each point below and will revise the manuscript to incorporate these improvements while preserving the elementary character of the argument.

read point-by-point responses

Referee: The manuscript does not state the final explicit bound on the distance (e.g., total-variation or P(X≠Y)) obtained from the coupling; without this statement the claim of a 'much more general result' cannot be evaluated against standard bounds such as Le Cam's inequality.

Authors: We agree that an explicit statement of the bound is necessary for direct comparison with classical results. The coupling yields the inequality P(X ≠ Y) ≤ n(1 − e^{-p} − p e^{-p}) ≤ (n p^2)/2, and therefore the total-variation distance satisfies d_TV(Law(X), Law(Y)) ≤ (n p^2)/2. We will add a clearly labeled theorem stating this bound together with a short remark comparing it to Le Cam's inequality. revision: yes
Referee: The joint construction of the coupling (X binomial, Y Poisson) must be written out with the precise definition of the auxiliary variables and the verification that the marginals are exactly Bin(n,p) and Pois(np); the current description leaves open whether any measurability or independence step requires tools beyond the elementary facts invoked.

Authors: The construction is as follows: let X_1, …, X_n be i.i.d. Bernoulli(p) random variables and let Y_1, …, Y_n be i.i.d. Poisson(p) random variables, with the two families mutually independent. For each i we realize the pair (X_i, Y_i) on a common probability space so that the marginals are exactly Bernoulli(p) and Poisson(p) respectively (the explicit joint law is the one that attains the bound 1 − e^{-p} − p e^{-p} on the disagreement probability). We then set X = sum X_i and Y = sum Y_i. The marginal of X is Bin(n,p) by the usual sum of independent Bernoullis; the marginal of Y is Pois(np) because the sum of independent Poisson(p) variables is Poisson(np). All steps use only the definitions of independence, finite sums, and the Poisson and Bernoulli distributions; no measurability or limit arguments are required. We will expand the manuscript with this level of detail and an explicit verification of the marginals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs an explicit coupling between Bin(n,p) and Pois(np) by representing the binomial as a sum of independent Bernoullis and introducing independent Poisson(p) variables, then bounding P(X ≠ Y) via the elementary inequality 1 − e^{-p} − p e^{-p} ≤ p²/2. This yields a non-asymptotic total-variation bound directly from the definitions of the distributions, independence, and finite sums. No self-citations, fitted parameters, ansatzes, or redefinitions of the target result appear; the argument relies solely on standard probability primitives and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard probability axioms and the existence of a coupling; no free parameters, invented entities, or ad-hoc assumptions are mentioned in the abstract.

axioms (1)

standard math Kolmogorov axioms of probability (or equivalent elementary rules for expectation and independence)
The note states it uses only elementary results from probability.

pith-pipeline@v0.9.0 · 5345 in / 1029 out tokens · 35177 ms · 2026-05-09T16:39:56.232086+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Durrett (1994)

R. Durrett (1994). The essentials of probability. The Duxbury Press

work page 1994
[2]

Lindvall (1992) Lectures on the Coupling Method

T. Lindvall (1992) Lectures on the Coupling Method. Wiley

work page 1992
[3]

Pitman (1993) Probability

J. Pitman (1993) Probability. Springer-Verlag

work page 1993
[4]

Roch (2024) Modern discrete probability : an essential toolkit

S. Roch (2024) Modern discrete probability : an essential toolkit. Cam- bridge University Press 3

work page 2024
[5]

R. B. Schinazi (2022) Probability with statistical applications (third edi- tion). Birkhauser. 4

work page 2022

[1] [1]

Durrett (1994)

R. Durrett (1994). The essentials of probability. The Duxbury Press

work page 1994

[2] [2]

Lindvall (1992) Lectures on the Coupling Method

T. Lindvall (1992) Lectures on the Coupling Method. Wiley

work page 1992

[3] [3]

Pitman (1993) Probability

J. Pitman (1993) Probability. Springer-Verlag

work page 1993

[4] [4]

Roch (2024) Modern discrete probability : an essential toolkit

S. Roch (2024) Modern discrete probability : an essential toolkit. Cam- bridge University Press 3

work page 2024

[5] [5]

R. B. Schinazi (2022) Probability with statistical applications (third edi- tion). Birkhauser. 4

work page 2022