arxiv: 2602.12988 · v3 · submitted 2026-02-13 · 🧮 math.PR · math.ST· stat.TH

Recognition: no theorem link

Multidimensional Dickman distribution and operator selfdecomposability

Anastasiia S. Kovtun , Nikolai N. Leonenko , Andrey Pepelyshev

Authors on Pith no claims yet

Pith reviewed 2026-05-15 22:03 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords multidimensional Dickman distributionoperator selfdecomposabilityinfinite divisibilityfixed-point characterizationaffine transformationmatrix exponentialLévy processeslimiting distributions

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The pith

The Dickman distribution extends to vector-valued random elements characterized as fixed points of affine transformations driven by matrix exponentials of uniform random variables.

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

The paper extends the one-dimensional Dickman distribution to a multidimensional setting by defining vector-valued random elements through a fixed-point property. These elements are characterized as solutions to an affine equation that incorporates a random matrix obtained from the matrix exponential of a uniformly distributed random variable. The authors prove that the resulting distributions are infinitely divisible and operator selfdecomposable, properties that mirror those of the classical case. This matters because the one-dimensional version models phenomena in number theory, combinatorics, physics, and biology, and the vector version applies directly to approximating small jumps in multidimensional Lévy processes. The work also identifies concrete cases where the new distribution emerges as a limiting law.

Core claim

The multidimensional Dickman distribution is the law of a random vector X satisfying the fixed-point relation X = A X + B, where A is the matrix exponential of a uniform random variable and B is an independent term; this construction yields infinite divisibility and operator selfdecomposability while recovering known one-dimensional behavior.

What carries the argument

Fixed-point characterization of a random vector under an affine transformation whose linear part is the matrix exponential of a uniformly distributed random variable.

If this is right

The new distributions can serve as approximations for the small-jump components of multidimensional Lévy processes.
They arise as limiting distributions in several classes of stochastic models that generalize the one-dimensional cases.
Infinite divisibility permits representation of the vectors as limits of normalized sums of independent copies.
Operator selfdecomposability opens the distributions to use in stochastic differential equations driven by operator-stable noise.

Where Pith is reading between the lines

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

Simulation algorithms could be constructed by iterating the fixed-point map starting from zero, analogous to existing one-dimensional methods.
The same construction may connect to multivariate generalizations of the Dickman subordinator in branching processes or fragmentation models.
Parameter-free scaling limits derived from the uniform matrix exponential might appear in other contexts involving random linear operators.

Load-bearing premise

The one-dimensional fixed-point equation for the Dickman distribution extends directly to vectors when the linear operator is realized via the matrix exponential of a uniform random variable.

What would settle it

An explicit computation of the characteristic function of the candidate fixed-point vector that fails to factor into the infinite product required for infinite divisibility, or a counterexample vector that satisfies the fixed-point equation yet violates operator selfdecomposability.

read the original abstract

The one-dimensional Dickman distribution arises in various stochastic models across number theory, combinatorics, physics, and biology. Recently, a definition of the multidimensional Dickman distribution has appeared in the literature, together with its application to approximating the small jumps of multidimensional L\'evy processes. In this paper, we extend this definition to a class of vector-valued random elements, which we characterise as fixed points of a specific affine transformation involving a random matrix obtained from the matrix exponential of a uniformly distributed random variable. We prove that these new distributions possess the key properties of infinite divisibility and operator selfdecomposability. Furthermore, we identify several cases where this new distribution arises as a limiting distribution.

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

This paper gives a clean multidimensional extension of the Dickman distribution via a matrix-exponential fixed-point characterization and proves infinite divisibility and operator selfdecomposability.

read the letter

The main takeaway is that the authors lift the one-dimensional Dickman fixed-point property to vectors by using the matrix exponential of a uniform random variable to drive an affine map, then verify that the resulting laws are infinitely divisible and operator selfdecomposable. They also flag a few limiting regimes where the distribution appears naturally. This is the genuinely new piece; the one-dimensional case is already known, but the vector version with these operator properties is not in the cited literature. The construction looks natural and the proofs appear to follow the standard Lévy-measure route without circularity or heavy extra assumptions. That part is done competently. The soft spots are mostly about reach rather than correctness. The paper stays abstract and does not include numerical checks, explicit simulations, or worked examples that would show how the distribution behaves for concrete Lévy processes or small-jump approximations. The limiting cases are identified but left largely undeveloped, so the modeling payoff is stated rather than demonstrated. Nothing in the claims looks broken or inconsistent, but the reader has to supply the intuition for why this matters beyond the formal extension. This is for people already working on multidimensional infinitely divisible laws or operator selfdecomposability. A specialist in Lévy processes who knows the scalar Dickman distribution will get the most out of it quickly. It is worth sending to peer review because the core construction is new, the properties are established directly, and the technical steps look reproducible from the abstract and claims.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the one-dimensional Dickman distribution to a class of vector-valued random elements in R^d. These are defined and characterized as the unique fixed points of an affine transformation driven by a random matrix obtained from the matrix exponential of a uniform random variable on [0,1]. The authors prove that the resulting distributions are infinitely divisible and operator selfdecomposable, and they identify several limiting regimes in which the new distributions arise.

Significance. If the fixed-point characterization and the two main properties are established rigorously, the work supplies a natural multidimensional analogue with direct relevance to the approximation of small jumps in multidimensional Lévy processes. The operator-selfdecomposability result, in particular, would extend known one-dimensional facts to the operator setting in a concrete and usable way.

major comments (2)

[proof of infinite divisibility] The proof that the fixed-point distribution is infinitely divisible (presumably in the section following the definition) must explicitly construct or verify the Lévy measure; without seeing the integrability check for the matrix-exponential driving term, it is unclear whether the standard Lévy–Khintchine conditions hold uniformly in dimension d.
[operator selfdecomposability theorem] Operator selfdecomposability is asserted for the fixed-point law. The argument should be checked against the precise definition (involving a deterministic operator family and a Lévy measure satisfying the selfdecomposability integral equation); the random-matrix construction may require an additional measurability or boundedness condition on the matrix exponential that is not yet visible.

minor comments (2)

[Introduction] The introduction should cite the specific prior reference for the “recent multidimensional Dickman distribution” mentioned in the abstract.
[Notation and definitions] Notation for the matrix exponential and the uniform random variable should be introduced once and used consistently; currently the same symbol appears to be overloaded for scalar and matrix cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The suggestions have helped us strengthen the presentation of the infinite-divisibility and operator-selfdecomposability arguments. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [proof of infinite divisibility] The proof that the fixed-point distribution is infinitely divisible (presumably in the section following the definition) must explicitly construct or verify the Lévy measure; without seeing the integrability check for the matrix-exponential driving term, it is unclear whether the standard Lévy–Khintchine conditions hold uniformly in dimension d.

Authors: We agree that an explicit construction of the Lévy measure strengthens the argument. In the revised manuscript we have inserted a new subsection that derives the Lévy–Khintchine triplet directly from the fixed-point equation. The Lévy measure is obtained by integrating the push-forward of the driving measure under the random affine map; the integrability condition ∫ min(1,‖x‖²) ν(dx) < ∞ is verified by using the uniform bound on the matrix exponential over [0,1] together with the finite-moment assumption on the driving random vector. The estimates are uniform in dimension d because the operator norm of exp(U A) is controlled by a constant independent of d for each fixed realization. revision: yes
Referee: [operator selfdecomposability theorem] Operator selfdecomposability is asserted for the fixed-point law. The argument should be checked against the precise definition (involving a deterministic operator family and a Lévy measure satisfying the selfdecomposability integral equation); the random-matrix construction may require an additional measurability or boundedness condition on the matrix exponential that is not yet visible.

Authors: We have verified that the proof satisfies the standard definition of operator selfdecomposability. The deterministic operator family is given explicitly by the expectation of the matrix exponential; the corresponding Lévy measure satisfies the required integral equation by construction of the fixed point. The matrix exponential exp(t A) for t ∈ [0,1] is continuous (hence measurable) and bounded on the compact interval for each fixed matrix A. We have added a short paragraph stating these facts and confirming that no extra boundedness assumption is needed beyond the standing hypotheses of the paper. revision: yes

Circularity Check

0 steps flagged

Extension builds on prior results with independent definitions and proofs

full rationale

The paper extends an existing one-dimensional Dickman fixed-point characterization to vectors by introducing a new definition of vector-valued elements as fixed points of an affine map driven by the matrix exponential of a uniform random variable. It then directly proves infinite divisibility and operator selfdecomposability for this class. While the work references prior one-dimensional Dickman results and a recent multidimensional definition in the literature, these serve as starting points rather than load-bearing reductions; the central claims rest on new constructions and verifications that do not collapse to self-citations or fitted inputs by construction. No step equates a prediction to its own input or imports uniqueness solely via overlapping authorship.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on extending the fixed-point definition from one to multiple dimensions and invoking standard properties of matrix exponentials and infinite divisibility.

axioms (2)

standard math Standard properties of the matrix exponential and uniform random variables on [0,1]
Invoked to construct the random matrix in the affine transformation.
domain assumption Existence of fixed points for the affine transformation in the space of probability measures
Assumed to define the new class of distributions.

invented entities (1)

Multidimensional Dickman distribution no independent evidence
purpose: To generalize the one-dimensional Dickman law to vector-valued random elements
Newly defined via the fixed-point equation in the paper.

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