arxiv: 2603.08598 · v2 · submitted 2026-03-09 · 🧮 math.PR

Recognition: 1 theorem link

· Lean Theorem

Asymptotic formulas for products of Poisson distributions

D\v{z}iugas Chvoinikov , Jonas \v{S}iaulys

Authors on Pith no claims yet

Pith reviewed 2026-05-15 13:25 UTC · model grok-4.3

classification 🧮 math.PR

keywords asymptotic tail probabilityPoisson productLaplace methodconstrained saddle-pointLambert functionlarge deviationsStirling approximation

0 comments

The pith

The tail probability for the product of fixed-parameter independent Poissons admits an explicit Laplace-type asymptotic with O(log n) remainder in the exponent.

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

The paper studies the probability that the product of N independent Poisson random variables with fixed positive parameters exceeds a growing threshold n. It constructs an explicit approximation to this tail using Stirling's formula on the Poisson probabilities, a constrained saddle-point analysis of the joint generating function, and the Lambert function to locate the saddle. The resulting formula includes a Gaussian prefactor whose width is evaluated under the product constraint, leaving an error of order log n inside the exponent. A sympathetic reader would care because the approximation gives concrete numerical access to rare-event probabilities in multiplicative counting processes without requiring Monte Carlo or exact convolution.

Core claim

We derive a refined Laplace-type asymptotic formula for the tail probability P(ξ1⋯ξN≥n), based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the constrained Gaussian prefactor. This yields an explicit approximation with an O(log n) remainder term in the exponent.

What carries the argument

The constrained saddle-point method applied to the cumulant generating function of the log-product, with the saddle location obtained explicitly via the Lambert function, which then supplies both the exponential rate and the Gaussian prefactor under the product constraint.

If this is right

The tail probability admits an explicit closed-form approximation usable for direct numerical evaluation at large n.
The error term permits controlled approximation in applications involving rare multiplicative events.
The same saddle-point location governs the most likely configuration of the Poisson counts that realize the product threshold.
The approximation is uniform over bounded ranges of the fixed parameters.

Where Pith is reading between the lines

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

The same saddle-point technique could be tested on products of other lattice distributions whose generating functions admit similar analytic continuations.
Direct numerical comparison for moderate n would reveal whether the O(log n) remainder is already useful in practice or requires further correction terms.
The result suggests a route to large-deviation principles for log-products of independent discrete random variables.

Load-bearing premise

The Poisson parameters are held fixed and positive while the threshold n tends to infinity.

What would settle it

For concrete fixed λ values, compute the exact tail probability numerically for a sequence of large n and verify whether the ratio of the explicit formula to the true probability tends to one at the rate consistent with an O(log n) error inside the exponent.

Figures

Figures reproduced from arXiv: 2603.08598 by D\v{z}iugas Chvoinikov, Jonas \v{S}iaulys.

**Figure 1.** Figure 1: Comparison of the Monte Carlo values of − log P (3) n with the explicit part of the logarithmic asymptotic formula for independent random variables ξ1 ∼ Poisson(1), ξ2 ∼ Poisson(2), and ξ3 ∼ Poisson(3). The figure illustrates the leading decay trend. References [1] Haight, F.A. Handbook of the Poisson Distribution; John Wiley & Sons: New York, 1967. [von Mises(1921)] Von Mises, R. Uber die wahrscheinlichk… view at source ↗

read the original abstract

In this paper, we study the asymptotic behaviour of the product tail probability $ \mathbb{P}(\xi_1\cdots\xi_N \geqslant n), $ where $\{\xi_1,\ldots,\xi_N\}$ is a finite collection of independent Poisson random variables with positive parameters $\lambda_1,\ldots,\lambda_N$. We derive a refined Laplace-type asymptotic formula for the tail probability, based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the constrained Gaussian prefactor. This yields an explicit approximation with an $O(\log n)$ remainder term in the exponent.

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

The paper gives a usable refined asymptotic for the tail of the product of fixed independent Poissons, obtained via constrained saddle-point and Lambert W with an explicit O(log n) remainder.

read the letter

The main thing here is a concrete asymptotic formula for P(∏ ξ_i ≥ n) where the ξ_i are independent Poissons with fixed positive rates λ_i and n → ∞. They minimize the sum of the usual rate functions I(x_i) = x_i log(x_i/λ_i) - x_i + λ_i subject to the product constraint, solve the Lagrange system with the Lambert function, and then extract the Gaussian prefactor while keeping an O(log n) term in the exponent. This is a direct application of Stirling plus saddle-point methods to this exact multiplicative tail setting. The combination with the explicit remainder and the constrained prefactor evaluation does not appear in the earlier references cited in the abstract, so the refinement is new for this problem. The stress-test note confirms that the multiplier produces a positive argument for the principal Lambert branch, so the saddle point is well-defined and there is no branch ambiguity. That part of the argument looks solid and uses only classical tools without circularity or invented entities. The soft spots are limited. The O(log n) remainder is explicit but not especially sharp, and the paper would be stronger if it included numerical checks for moderate n to show how the approximation behaves in practice. The regime is restricted to fixed λ_i and fixed N, which is stated up front and matches the title, so it is not a hidden limitation. This is aimed at specialists in asymptotic probability and large deviations who need tail estimates for products of discrete random variables. A reader working on similar constrained saddle-point problems would get a usable formula from it. The work shows clear thinking and straightforward engagement with the standard literature, so it deserves a serious referee. I would recommend sending it to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript derives a refined Laplace-type asymptotic formula for the tail probability P(ξ1 ⋯ ξN ≥ n) where the ξi are independent Poisson random variables with fixed positive parameters λ1, …, λN and n → ∞. The derivation combines Stirling's logarithmic approximation, a constrained saddle-point method applied to the sum of rate functions I(xi) = xi log(xi/λi) − xi + λi subject to the product constraint, the Lambert W function to solve the resulting saddle equations, and an explicit evaluation of the constrained Gaussian prefactor, yielding an approximation whose error is O(log n) in the exponent.

Significance. If the central derivation is correct, the result supplies an explicit, usable asymptotic expression for a multiplicative tail that improves upon standard large-deviation bounds by including the prefactor and a controlled remainder. The approach relies entirely on classical, parameter-free tools (Stirling, saddle-point, principal branch of the Lambert W function) whose validity is independent of the target statement, and the uniqueness of the positive saddle point is confirmed by the sign of the Lagrange multiplier in the tail regime.

major comments (2)

[Abstract and §4] Abstract and §4 (Theorem on the asymptotic formula): the O(log n) remainder in the exponent is asserted but the manuscript provides neither a complete expansion of the error terms arising from the saddle-point approximation and the Gaussian integral nor any numerical verification that the claimed order is attained; this error control is load-bearing for the claim of a 'refined' formula.
[§3] §3 (constrained saddle-point equations): while the reduction to w_j exp(w_j) = −μ/λ_j is standard, the manuscript does not explicitly verify that the Hessian of the constrained rate function remains positive definite uniformly in the large-n regime, which is required to justify the Gaussian prefactor without additional logarithmic corrections.

minor comments (3)

[§2] The rate function I(x) should be written out in full at the beginning of §2 rather than introduced only inside the saddle-point analysis.
[§3] Notation for the Lagrange multiplier μ and the solution w_j should be introduced once and used consistently; the current alternation between μ and the implicit multiplier is mildly confusing.
[Introduction] A short paragraph comparing the new formula with the classical Chernoff or crude large-deviation bound would help readers gauge the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments on error control and the Hessian. We have revised the manuscript to strengthen these aspects while preserving the original derivation.

read point-by-point responses

Referee: [Abstract and §4] Abstract and §4 (Theorem on the asymptotic formula): the O(log n) remainder in the exponent is asserted but the manuscript provides neither a complete expansion of the error terms arising from the saddle-point approximation and the Gaussian integral nor any numerical verification that the claimed order is attained; this error control is load-bearing for the claim of a 'refined' formula.

Authors: We agree that the original error discussion was implicit. In the revised version we have added an appendix that expands the remainder explicitly: the Stirling contribution is O(log n) by the standard logarithmic form, the saddle-point contour shift contributes o(1) once the uniform Hessian bound is in place, and the Gaussian integral error is absorbed into the same O(log n) term. We have also inserted a short numerical section in §4 that compares the asymptotic expression against direct summation for N=2,3 and n up to 10^4, confirming that the observed error stays within the claimed order. revision: yes
Referee: [§3] §3 (constrained saddle-point equations): while the reduction to w_j exp(w_j) = −μ/λ_j is standard, the manuscript does not explicitly verify that the Hessian of the constrained rate function remains positive definite uniformly in the large-n regime, which is required to justify the Gaussian prefactor without additional logarithmic corrections.

Authors: We thank the referee for highlighting this gap. The revised §3 now contains a new lemma that proves uniform positive-definiteness of the constrained Hessian for all sufficiently large n. The argument uses the explicit form of the saddle point via the principal Lambert W branch, shows that each diagonal entry of the Hessian is bounded below by a positive constant independent of n (because μ remains negative and bounded away from zero in the tail regime), and verifies that the off-diagonal Lagrange-multiplier term does not destroy definiteness. This bound is then used to control the Gaussian prefactor without extra logarithmic factors. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's derivation of the refined Laplace-type asymptotic for the product tail probability relies on Stirling's approximation, the constrained saddle-point method applied to the sum of rate functions I(x_i), and the Lambert W function to solve the resulting transcendental equation for the Lagrange multiplier. These are classical, externally validated mathematical tools whose correctness does not depend on the target tail probability or any fitted parameters from the present work. The uniqueness of the positive real solution for the principal branch W_0 follows directly from the sign of the argument when μ < 0 in the large-n regime, without self-citation chains, ansatz smuggling, or renaming of known results. The Gaussian prefactor evaluation proceeds from the Hessian at this saddle point, again using standard local analysis. No load-bearing step reduces to a definition or input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard analytic tools without additional free parameters fitted to data or new postulated entities.

axioms (2)

standard math Stirling's logarithmic approximation is valid for large positive arguments
Invoked to approximate the Poisson probabilities inside the tail integral.
domain assumption The constrained saddle-point equation admits a unique positive solution expressible via the Lambert function
Required for the explicit leading-term location when n is large.

pith-pipeline@v0.9.0 · 5400 in / 1187 out tokens · 48231 ms · 2026-05-15T13:25:55.625963+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

saddle-point equations ... k_i^* exp{α n + 1/2 k_i^*} = λ_i ... solved by Lambert W

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

[von Mises(1921)] Von Mises, R

Haight, F.A.Handbook of the Poisson Distribution; John Wiley&Sons: New York, 1967. [von Mises(1921)] Von Mises, R. Uber die wahrscheinlichkeit seltener ereignisse.Z. Angew. Math. Mech.1921,1, 121-124. [Feller (1950)] Feller, W.Introduction to Probability Theory and Some of its Applications; John Wiley and Sons Inc.: New York; Chapman and Hall Limited: Lon...

work page 1967
[2]

Bernoulli trials with variable probabilities.Lith

Mačys, J.J. Bernoulli trials with variable probabilities.Lith. Math. J.1979,19, 533-537

work page 1979
[3]

In New Trends in Probability and Statis- tics, Vol

Šiaulys,J.The von Mises theorem in number theory. In New Trends in Probability and Statis- tics, Vol. 2, Analytic and Probabilistic Methods in Number Theory, F. Schweiger and E. Manstavičius (eds.; VSF, Utrecht/TEV:Vilnius, Lithuania, 1992, pp. 293-310

work page 1992
[4]

Convergence to the Poisson law

Šiaulys, J. Convergence to the Poisson law. III. Method of moments.Lit. Math. J.1998,38, 374–390. https://doi.org/10.1007/BF02465821

work page doi:10.1007/bf02465821 1998
[5]

Poisson approximations for telecommunications networks.J

Brown, T.C.; Pollet, P.K. Poisson approximations for telecommunications networks.J. Aus- tral. Math. Soc. Ser B1991,32, 348-364

work page
[6]

3; Oxford University Press: Oxford, 1992

Kingman, J.F.C.Poisson Processes, vol. 3; Oxford University Press: Oxford, 1992

work page 1992
[7]

1, second ed.; Springer: New York, 2003

Daley, D.J.; Vere-Jones, D.An Introduction to the Theory of Point Processes, vol. 1, second ed.; Springer: New York, 2003

work page 2003
[8]

A primer on special modeling and analysis in wireless networks.IEEE Commun

Andrews, J.G.; Gonti, R.K.; Haenggi, M.; Jindal, N.; Weber, S. A primer on special modeling and analysis in wireless networks.IEEE Commun. Mag.2010,48, 156-163

work page 2010
[9]

A Poisson process- based random access chanel for 5G and beyond netwoks.Mathematics2021,9, 508

Almagrabi, A.O.; Ali, R.; Alghazzawi, D.; AlBarakati, A.; Khurshaid, T. A Poisson process- based random access chanel for 5G and beyond netwoks.Mathematics2021,9, 508

work page
[10]

In: Staffa, M.,Cabibihan,J.J.,Siciliano,B.,Ge,S.S.,Bodenhagen,L.,Tapus,A.,Rossi,S.,Cav- allo, F., Fiorini, L., Matarese, M., He, H

Ben-Haim, Y.Robust reliability and the Poisson process.In: RobustReliabilityintheMechan- ical Sciences; Springer: Berlin, Heidelberg, 1996, pp. 189-203, https://doi.org/10.1007/978- 3-642-61154-4−8

work page doi:10.1007/978- 1996
[11]

In: Stochastic Processes

Nakagawa, T.Poisson processes. In: Stochastic Processes. Springer Series in Reliability En- gineering; Springer: London, 2011, pp. 7-46, https://doi.org/10.1007/978-0-85729-274-2−2

work page doi:10.1007/978-0-85729-274-2 2011
[12]

Extreme shock model with change point based on the Poisson process shocks.Appl

Goyal, D.X.; Xie, M. Extreme shock model with change point based on the Poisson process shocks.Appl. Stoch. Models Bus. Ind.2024,40, 1635-1650

work page 2024
[13]

Confidence bounds for compound Poisson process.Stat

Skarupski, M.; Wu, Q.H. Confidence bounds for compound Poisson process.Stat. Pap.2024, 65, 5351-5377

work page 2024
[14]

Variant Poisson item count technique with non-compilance.Mathematics2025,13,2973

Tang, M.L.; Wu, Q.; Chow, D.H.S.; Tian, G.L. Variant Poisson item count technique with non-compilance.Mathematics2025,13,2973

work page
[15]

A novel lifetime analysis of repairable systems via Daubechies wavelets.Ann

Wu, J.C.; Dohi, T.; Okamura, H. A novel lifetime analysis of repairable systems via Daubechies wavelets.Ann. Oper. Res.2025,349, 287-314

work page 2025
[16]

In: Zhao, Q.Q., Chung, I.H., Zheng, J., Kim, J

Cha, J.H.Generalizations of the Poisson Process and their reliability applications. In: Zhao, Q.Q., Chung, I.H., Zheng, J., Kim, J. (eds) Reliability Analysis and Maintenance Optimiza- tion of Complex Systems. Springer Series in Reliability Engineering; Springer: Cham, 2025, pp. 85-95, https://doi.org/10.1007/978-3-031-70288-4−6

work page doi:10.1007/978-3-031-70288-4 2025
[17]

Simulating a Poisson Cluster Process for internet traffic pocket arrivals.Comput

González-Arévalo, B.; Ray, J. Simulating a Poisson Cluster Process for internet traffic pocket arrivals.Comput. Commun.2010,33, 512-618

work page 2010
[18]

Small and large scale behaviour of moments of Poisson cluster processes.ESAIM - Probab

Antures, N.; Pipiras, V.; Abey, P.; Veitch, D. Small and large scale behaviour of moments of Poisson cluster processes.ESAIM - Probab. Stat.2017,21, 369-393, https://doi.org/10.1051/ps/2017018. 25

work page doi:10.1051/ps/2017018 2017
[19]

Markov-modulated Poisson process modeling for machine-to-machine heterogeneous traffic.Appl

El Fawal, A.H.; Mansour, A.; Nasser, A. Markov-modulated Poisson process modeling for machine-to-machine heterogeneous traffic.Appl. Sci.2024,14, 8561, https://doi.org/10.3390/app14188561

work page doi:10.3390/app14188561 2024
[20]

The Poisson extension of the unrelated question model: improving surveys with time-constrained questions on sensitive topics

Iberl, B.; Aljovic, A.; Ulrich, R.; Reiber, F. The Poisson extension of the unrelated question model: improving surveys with time-constrained questions on sensitive topics. |Surv. Res. Methods2024,18, 21–38. https://doi.org/10.18148/srm/2024.v18i1.8252

work page doi:10.18148/srm/2024.v18i1.8252 2024
[21]

Large and moderate deviations in Poisson navigations

Ghosh, P.P.; Jahnel, B.; Jhawar, S.K. Large and moderate deviations in Poisson navigations. Adv. Appl. Probab.Published online 2025:1-38, https://doi:10.1017/apr.2025.10025

work page doi:10.1017/apr.2025.10025 2025
[22]

On the decomposition of Poisson laws.Dokl

Raikov, D. On the decomposition of Poisson laws.Dokl. Acad. Sci. URSS1937,4, 9-14

work page
[23]

On the decomposition of Gaus and Poisson laws.Bull

Raikov, D. On the decomposition of Gaus and Poisson laws.Bull. Acad. Sci. URSS Ser. Math.1938,1, 91-124

work page 1938
[24]

Foos, S.; Korshunov, D.; zachary, S.An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd ed; Springer: New York, 2013

work page 2013
[25]

Nair, J.; Wierman, A.; Zwart, B.The Fundamentals of Heavy Tails: Properties, Emergence, and Estimation; Cambridge University Press: Cambridge, 2022

work page 2022
[26]

Mitzenmacher, M.; Upfal, E.Probability and Computing: Randomized Algorithms and Prob- abilistic Analysis; Cambridge University Press: Cambridge, UK, 2005

work page 2005
[27]

Wong, R.Asymptotic Approximations of Integrals; SIAM: Philadelphia, USA, 2001

work page 2001
[28]

On the remarkable properties of a series of Lambert and other (in Latin)BActa Acad

Euler, L. On the remarkable properties of a series of Lambert and other (in Latin)BActa Acad. Sci. Imp. Petrop.1783,II, 29-51

work page
[29]

On the LambertW function.Adv

Corless, R.M.; Gonett, G.H.; Hare, D.E.G.;Jeffrey, D.H.; Knuth, D.E. On the LambertW function.Adv. Comput. Math.1996,5, 329-359. 26

work page 1996