Mod-Poisson Convergence

• Mod-Poisson convergence is a framework for integer-valued random variables whose primary Poisson behavior is corrected by an analytic residue capturing second-order fluctuations.
• It employs deconvolution via Fourier and Laplace transforms to isolate factorial cumulants, allowing precise approximation schemes and quantification of error rates in discrete models.
• The approach underpins diverse applications—from combinatorial and arithmetic analyses to risk modeling—often revealing a splitting phenomenon with a universal Gamma correction factor.

Mod-Poisson convergence is a framework for integer-valued random variables whose first-order behavior is well approximated by a Poisson law, but whose Laplace or Fourier transforms retain a non-trivial residue after Poisson renormalization. One studies sequences $(X_n)$ and speeds $(t_n)$ such that the normalized transforms converge locally uniformly to an analytic limiting function $\Psi$, which records second-order fluctuations, factorial-cumulant corrections, and model-specific structure beyond the leading Poisson term. In discrete lattice problems, mod-Poisson is a principal instance of mod-$\phi$ convergence and supports refined local asymptotics, large and moderate deviation estimates, signed and positive approximation schemes, and, in several arithmetic and combinatorial models, a splitting phenomenon in which the limiting residue factors into a universal Gamma term and an independent-model correction (Barhoumi-Andréani, 22 Jul 2025, Chhaibi et al., 2015).

1. Definition and analytical setting

Let $(X_n)$ be integer-valued random variables and let $(t_n)$ be a sequence of positive numbers. On the Laplace side, mod-Poisson convergence at speed $(t_n)$ with limiting analytic function $\Psi$ means that, for $z$ in a domain containing $1$,

$(t_n)$0

locally uniformly. Since $(t_n)$1, this is equivalently

$(t_n)$2

On the Fourier side, with $(t_n)$3,

$(t_n)$4

locally uniformly for $(t_n)$5 near $(t_n)$6. In the Laplace formulation used in the 2025 analysis of the splitting phenomenon, the limit satisfies $(t_n)$7 and the conjugation symmetry $(t_n)$8; moreover, $(t_n)$9 is defined only up to multiplication by an exponential factor $\Psi$0, because shifting the speed by $\Psi$1 changes the normalization by exactly such a term (Barhoumi-Andréani, 22 Jul 2025).

Within mod-$\Psi$2 convergence, one fixes an infinitely divisible reference law with Lévy–Khintchine exponent $\Psi$3 and studies

$\Psi$4

Mod-Poisson is the specialization $\Psi$5, and can equally be formulated in Laplace form through

$\Psi$6

for $\Psi$7 in a neighborhood of $\Psi$8. This places Poisson normalization on the same conceptual footing as mod-Gaussian normalization, while preserving the discrete, lattice-valued character of the underlying laws (Méliot et al., 2022).

2. Residues, cumulants, and the discrete harmonic-analytic viewpoint

The residue function is the stabilized deconvolution of $\Psi$9 by the Poisson reference law. On the Fourier side it is

$\phi$0

and on the Laplace side

$\phi$1

Its role is cumulant-theoretic: Poisson normalization removes the leading Poisson cumulants, and the residue encodes the remaining second-order fluctuations. In the Bernoulli-sum model $\phi$2 with $\phi$3 independent and $\phi$4, one has

$\phi$5

where $\phi$6. After Poisson deconvolution, the $\phi$7 term disappears and

$\phi$8

Expanding in powers of $\phi$9 gives

$(X_n)$0

with

$(X_n)$1

The coefficients satisfy the Cauchy-type bound

$(X_n)$2

which is the basic regularity input for higher-order approximation theory (Méliot et al., 2022).

In the discrete setting, the natural analytic environment is the Wiener algebra of absolutely convergent Fourier series. If $(X_n)$3 is a signed measure on $(X_n)$4, then

$(X_n)$5

so Wiener norms are exactly total-variation norms. This makes deconvolution by a reference infinitely divisible law legitimate through Wiener's theorem, and it underlies the transfer of Fourier estimates on residues to asymptotic equivalents in local, Kolmogorov, and total variation distances. The same framework extends to $(X_n)$6, where the asymptotic constants reveal correlations between components even when the reference compound Poisson law factorizes coordinatewise (Chhaibi et al., 2015).

3. Splitting, auxiliary randomisation, and the universal Gamma factor

A central structural development is the splitting phenomenon: after Poisson renormalization, the limiting function often factors as

$(X_n)$7

where $(X_n)$8 is a model-dependent factor computed from an independent approximation and $(X_n)$9 is universal. The explicit Euler product is

$(t_n)$0

with $(t_n)$1 Euler’s constant. In the permutation model this is exactly the mod-Poisson limit for the total number of cycles; in arithmetic and function-field models it appears as a correction to an independent prime or irreducible-factor approximation (Barhoumi-Andréani, 22 Jul 2025).

The common mechanism is an auxiliary randomisation that reveals an independence structure hidden by conditioning on size. For uniform permutations, geometric randomisation yields independent Poisson cycle counts: $(t_n)$2 For integers, the 2025 analysis introduces the delta-Zeta distribution $(t_n)$3 on $(t_n)$4,

$(t_n)$5

and proves the randomisation identity

$(t_n)$6

Khintchin’s theorem gives independent $(t_n)$7-adic valuations for $(t_n)$8, while the subordinated parameter has exponential fluctuations: $(t_n)$9 Since

$(t_n)$0

the Mellin transform of the auxiliary randomisation is exactly what produces the universal Gamma factor (Barhoumi-Andréani, 22 Jul 2025).

A closely related viewpoint was already visible in the penalised model for the Sathe–Selberg theorem, where the limiting function for $(t_n)$1 was decomposed as

$(t_n)$2

with

$(t_n)$3

and

$(t_n)$4

This decomposition already isolates an arithmetic independent-prime term and a cycle/Gamma correction, and the later auxiliary-randomisation theory explains why such a factorization recurs across apparently unrelated models (Barhoumi-Andréani, 2017).

4. Canonical models and explicit limiting functions

The standard examples are combinatorial, arithmetic, and finite-field models in which the mean scale grows like $(t_n)$5 or $(t_n)$6 and the residue is explicit.

Model	Poisson speed	Limiting structure
Uniform permutations $(t_n)$7	$(t_n)$8	$(t_n)$9
Distinct prime divisors $\Psi$0	$\Psi$1	$\Psi$2
Prime divisors with multiplicity $\Psi$3	$\Psi$4	$\Psi$5
Irreducible factors of $\Psi$6	$\Psi$7	$\Psi$8
Jordan blocks in $\Psi$9	$z$0	$z$1

In the Sathe–Selberg case, the residue separates into a universal Gamma term and an independent-model factor obtained by replacing dependent valuations by independent Bernoulli or geometric variables. The structural theorem stated in the 2025 analysis shows that if mod-Poisson convergence for $z$2 is known at speed $z$3 with remainder $z$4 or $z$5, then necessarily

$z$6

For permutations, by contrast, the independent model is already exact enough that the limit is just $z$7 (Barhoumi-Andréani, 22 Jul 2025).

The class of examples is broader than the five models listed in the table. Weighted random permutations under measures

$z$8

exhibit mod-Poisson convergence when the associated logarithmic singularity has the form

$z$9

with parameters

$1$0

The Ewens measure is the special case $1$1. For random monic polynomials over $1$2, the number of distinct irreducible divisors $1$3 is mod-Poisson with

$1$4

and limiting alphabet

$1$5

These examples show that the residue can often be encoded by symmetric-function data or “alphabets” whose power sums record prime, cycle, or irreducible-factor contributions (Méliot et al., 2022).

5. Approximation schemes, Poisson–Charlier corrections, and quantitative bounds

Mod-Poisson convergence is not only asymptotic classification; it also yields constructive approximation schemes. If

$1$6

the order-$1$7 approximation is the signed measure $1$8 with Fourier transform

$1$9

For $(t_n)$00 this becomes the explicit Poisson–Charlier correction

$(t_n)$01

where $(t_n)$02 is $(t_n)$03. Expectations under $(t_n)$04 can be computed by forward differences: $(t_n)$05 Although $(t_n)$06 need not be positive, a monotone mass-reallocation converts it into a positive probability measure $(t_n)$07 without increasing the total variation error (Méliot et al., 2022).

The higher-order Chen–Stein bounds in the Bernoulli setting are explicit. If the residue coefficients satisfy

$(t_n)$08

and $(t_n)$09, then there exist universal constants $(t_n)$10 and $(t_n)$11 such that

$(t_n)$12

with $(t_n)$13 and $(t_n)$14. For $(t_n)$15, $(t_n)$16, and $(t_n)$17, if $(t_n)$18 then

$(t_n)$19

For dependent models with Bernoulli asymptotics, the same paper proves

$(t_n)$20

which yields $(t_n)$21 under sufficiently fast residue convergence (Méliot et al., 2022).

A different but complementary quantitative theory is based on Fourier analysis in the Wiener algebra. There, signed approximations aligned with the residue yield asymptotic equivalents in local, Kolmogorov, and total variation distances. Under the local expansion hypothesis

$(t_n)$22

one obtains one-dimensional asymptotics

$(t_n)$23

and

$(t_n)$24

Under the stronger derivative condition (H2), the total variation distance satisfies

$(t_n)$25

These formulas make the Hermite-function structure of the error explicit, and the same method extends to multidimensional compound-Poisson references (Chhaibi et al., 2015).

6. Relation to mod-$(t_n)$26, applications, and conceptual boundaries

Mod-Poisson belongs to the broader mod-$(t_n)$27 framework, but its lattice character imposes distinctive analytic behavior. For stable reference laws, the “zone of control” machinery yields Berry–Esseen-type estimates in Kolmogorov distance because the reference law has a Lebesgue density and the factor $(t_n)$28 is integrable. This does not transfer directly to Poisson reference laws, which are lattice-valued and have no density. Accordingly, in the mod-Poisson setting the appropriate quantitative outputs are local limit theorems, pointwise probability asymptotics, and moderate deviation estimates on the integer lattice rather than Kolmogorov approximation to a continuous limit (Féray et al., 2017).

A common misconception is therefore to identify mod-Poisson convergence with ordinary Poisson approximation. The residue function carries the deviation from a pure Poisson law, and in all the approximation theories above it is the object that determines correction terms, matched factorial cumulants, and higher-order error rates. Another common misconception is to regard the Gamma factor as intrinsic to every mod-$(t_n)$29 regime. The 2025 splitting theory states the opposite: the universal $(t_n)$30 factor arises specifically when the auxiliary randomisation parameter has exponential fluctuations, whereas in mod-Gaussian settings the universal factor need not be Gamma and may instead be associated with Barnes $(t_n)$31 or random-matrix limits. The existence of a single paradigm covering both mod-Poisson and mod-Gaussian is presented as an open challenge (Barhoumi-Andréani, 22 Jul 2025).

The framework also has numerical applications. In factor credit portfolio models with conditionally independent defaults, the default count $(t_n)$32 is treated conditionally by mod-Poisson convergence when

$(t_n)$33

The resulting approximation schemes were used for the estimation of risk measures $(t_n)$34 and $(t_n)$35 and the computation of CDO tranche prices, and were compared with the recursive method, the large deviations approximation, the Chen–Stein method, and Monte Carlo simulation with and without importance sampling. The reported conclusion is that the mod-Poisson method leads to more accurate estimates while requiring less computational time. In these applications the practical algorithm consists of computing the conditional Poisson parameter $(t_n)$36, the power sums $(t_n)$37, the coefficients $(t_n)$38, and then evaluating corrected tails or call-type functionals before integrating over the common factor (Méliot et al., 2022).

The resulting picture is that mod-Poisson convergence is simultaneously an asymptotic language, a structural classification principle, and an approximation technology. It captures Poissonian first-order behavior together with non-Poisson residues; it explains recurring Gamma corrections through auxiliary randomisation and hidden independence; and it provides constructive schemes with explicit constants, especially for arithmetic and combinatorial lattice models.