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arxiv: 2604.14396 · v1 · submitted 2026-04-15 · 🧮 math.PR

On the tails of Dickman-like perpetuities

Alexander Iksanov , Oleh Iksanov This is my paper

Pith reviewed 2026-05-10 11:49 UTC · model grok-4.3

classification 🧮 math.PR
keywords perpetuitiesDickman distributiontail asymptoticsexponential change of measureprobability theorylarge deviations
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The pith

Perpetuities close to the Dickman distribution admit precise tail asymptotics via exponential change of measure.

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

This paper establishes the precise asymptotic behavior of the tails for perpetuities whose distributions are sufficiently close to the Dickman distribution. It employs a probabilistic approach using exponential change of measure to derive these asymptotics. A reader might care because such perpetuities model phenomena in insurance, finance, and branching processes where extreme values matter. The method provides clean expressions without dominating error terms by tilting the measure appropriately.

Core claim

By applying an exponential change of measure, the authors derive the precise tail asymptotic behavior for perpetuities with distributions close to the Dickman distribution.

What carries the argument

The exponential change of measure, which tilts the probability distribution exponentially to facilitate analysis of large deviations or tail events.

Load-bearing premise

The perpetuities' distributions must be close enough to the Dickman distribution so that the exponential change of measure captures the exact leading tail term without error terms taking over.

What would settle it

Compute or simulate the tail for a specific perpetuity close to Dickman and check if it matches the derived asymptotic formula; mismatch would falsify the precision claim.

read the original abstract

By using a probabilistic technique based on the exponential change of measure we find a precise tail asymptotic behavior of some perpetuities with distributions close to the Dickman 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.

Referee Report

1 major / 2 minor

Summary. The manuscript applies a probabilistic technique based on the exponential change of measure to derive precise tail asymptotic behavior for perpetuities whose distributions are close to the Dickman distribution.

Significance. If the derivation holds, the work provides a standard yet effective probabilistic tool for extracting tail asymptotics in fixed points of smoothing transforms near the Dickman law. This is relevant to branching processes, risk theory, and algorithmic analysis where such perpetuities arise. The approach of tilting the driving measure and verifying moment properties under the new measure aligns with existing literature on perpetuities and could extend to related classes if the closeness condition is controlled.

major comments (1)
  1. [Abstract and main result (likely Theorem 1 or Section 3)] The central claim depends on the perpetuities having distributions sufficiently close to the Dickman distribution so that the exponential change of measure produces precise asymptotics without dominating error terms. The abstract states this condition but does not specify the metric of closeness or bound the error; this needs explicit statement and verification in the main theorem and proof to support the precision claim.
minor comments (2)
  1. [Abstract] The abstract is concise but omits the precise form of the asymptotic (e.g., the leading constant or logarithmic correction). Consider adding a brief statement of the main result in the introduction for clarity.
  2. [Introduction and Section 2] Notation for the tilted measure and the closeness parameter should be introduced early and used consistently throughout the proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive major comment. We address it directly below.

read point-by-point responses

  1. Referee: [Abstract and main result (likely Theorem 1 or Section 3)] The central claim depends on the perpetuities having distributions sufficiently close to the Dickman distribution so that the exponential change of measure produces precise asymptotics without dominating error terms. The abstract states this condition but does not specify the metric of closeness or bound the error; this needs explicit statement and verification in the main theorem and proof to support the precision claim.

    Authors: We agree that the abstract could be more explicit on the notion of closeness. The manuscript defines Dickman-like perpetuities via a specific smallness condition on the driving measure (in the form of a bound on the difference of their Laplace transforms near the origin together with matching first moments), which is stated precisely in the hypothesis of the main theorem. Under this condition the proof already verifies that the tilted measure inherits the required integrability so that the error terms remain of strictly lower order. Nevertheless, to improve clarity we will revise the abstract to name the metric and add a short sentence on the resulting error control. We will also insert an explicit remark after the statement of the theorem that records the order of the remainder. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external probabilistic technique

full rationale

The paper's central claim applies the exponential change of measure—a standard, externally defined probabilistic tool—to obtain tail asymptotics for perpetuities close to the Dickman distribution. The abstract and high-level structure describe tilting the driving measure, verifying moment properties under the tilted law, and extracting the tail, without any equations or definitions that reduce the asymptotics to fitted inputs, self-referential constructions, or load-bearing self-citations. The method is presented as independent of the target result, consistent with known techniques for smoothing-transform fixed points. No self-definitional steps, renamed empirical patterns, or ansatz smuggling via citation are identifiable from the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, domain-specific axioms, or invented entities are identifiable. The work invokes the standard exponential change of measure technique from probability theory.

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discussion (0)

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Reference graph

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