Precise large deviations through a uniform Tauberian theorem

Gaia Pozzoli; Giampaolo Cristadoro

arxiv: 2407.04059 · v3 · pith:SBKV3F3Dnew · submitted 2024-07-04 · 🧮 math.PR · math-ph· math.MP

Precise large deviations through a uniform Tauberian theorem

Giampaolo Cristadoro , Gaia Pozzoli This is my paper

Pith reviewed 2026-05-23 23:13 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords large deviationsTauberian theoremstable distributionsregular variationLaplace-Stieltjes transformsrandom walksstopped sums

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The pith

A uniform Tauberian theorem for Laplace-Stieltjes transforms establishes large deviation principles for random variables attracted to spectrally positive stable distributions.

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

The paper proves a uniform version of the Tauberian theorem that converts Laplace-Stieltjes transform asymptotics into precise tail estimates, thereby obtaining a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable laws. This route works for processes where standard exponential-moment or concentration methods do not apply. A reader would care because the same device covers random walks with long-range memory kernels and randomly stopped sums whose stopping time is not concentrated or has infinite mean, all inside the framework of regular variation.

Core claim

By proving a uniform Tauberian theorem for Laplace-Stieltjes transforms we derive a large deviation principle for families of random variables belonging to the basin of attraction of spectrally positive stable distributions; the method applies directly to cases beyond the reach of existing techniques and reveals the role of the characteristic function when Cramér's condition fails.

What carries the argument

Uniform Tauberian theorem for Laplace-Stieltjes transforms, which supplies uniform asymptotic control on the tail behavior from transform data under regular variation.

If this is right

Large deviations hold for random walks whose increments have long-ranged memory kernels.
Large deviations hold for randomly stopped sums even when the stopping time has infinite mean or is not concentrated around its expectation.
The characteristic function governs the rate function once Cramér's condition is dropped.
All such results sit inside a single regular-variation framework.

Where Pith is reading between the lines

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

The same uniform Tauberian device may produce moderate-deviation principles or local limit theorems under only minor extra assumptions.
The approach could be tested on risk processes or storage models whose input has heavy tails and long memory.
It offers a route to large deviations for Markov chains or renewal processes whose return times are regularly varying.

Load-bearing premise

The random variables belong to the basin of attraction of spectrally positive stable distributions and obey the regular-variation conditions that make the uniform Tauberian theorem applicable.

What would settle it

An explicit family of random variables in the stable basin whose empirical large-deviation rate function differs from the one obtained via the uniform Tauberian route.

read the original abstract

We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature. Notable examples include large deviations for random walks with long-ranged memory kernels, as well as for randomly stopped sums where the random time $\mathrm N$ is either not concentrated around its expectation or has an infinite mean. The method reveals the role of the characteristic function when Cram\'er's condition is violated and provides a unified approach within regular variation.

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 proves a uniform Tauberian theorem for Laplace-Stieltjes transforms that yields LDPs for variables in the domain of attraction of spectrally positive stables and extends to long-memory walks plus randomly stopped sums with infinite mean.

read the letter

The core contribution is a uniform version of the Tauberian theorem that produces large deviation principles directly from the Laplace-Stieltjes transform under regular variation. This is positioned as a route to results for spectrally positive stable basins that also covers long-ranged memory kernels and randomly stopped sums where the stopping time is not concentrated or has infinite mean. The abstract frames this as going beyond standard techniques and as giving a unified treatment when Cramér's condition fails, with the characteristic function playing an explicit role. That combination of uniformity and the listed applications is the actual novelty claimed. The conditions are stated plainly as regular variation plus basin-of-attraction membership, with no sign of circularity or presupposed LDPs. The method is presented as straightforward to apply once the uniform theorem is in hand. The main limitation is that everything rests on the correctness and scope of the uniform Tauberian statement itself. Without the detailed proof steps it is impossible to check whether the uniformity carries through to the memory-kernel and infinite-mean cases without additional restrictions that might narrow the claimed advantage. The paper is a pure existence-and-application result in probability theory, so its value is tied to whether those steps hold. Readers working on large deviations for heavy-tailed or dependent processes would find the approach worth examining. It is coherent on its own terms and the claimed extension is specific enough to merit referee time rather than desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper claims to derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The approach is positioned as extending to long-memory random walks and randomly stopped sums (with stopping time N not concentrated around its mean or having infinite mean), while revealing the role of the characteristic function when Cramér's condition is violated and unifying results under regular variation.

Significance. If the uniform Tauberian theorem is established correctly, the result supplies a new method for precise LDPs in regimes beyond standard techniques, such as long-range dependence or infinite-mean stopping times. This could strengthen the toolkit for regular-variation-based large deviations in probability theory.

minor comments (2)

The abstract mentions applications to long-ranged memory kernels and randomly stopped sums but does not indicate where in the manuscript the corresponding theorems are stated; adding explicit theorem numbers or section references would improve readability.
Notation for the Laplace-Stieltjes transform and the uniform Tauberian statement could be introduced with a dedicated preliminary section to aid readers unfamiliar with the precise formulation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives an LDP for random variables in the basin of attraction of spectrally positive stable laws by first proving a uniform Tauberian theorem for Laplace-Stieltjes transforms under regular variation. This theorem is presented as an independent analytic result whose proof is the central contribution; the LDP then follows by direct application to the stated class of processes (long-memory walks, randomly stopped sums). No equation reduces the target LDP to a fitted parameter, no self-citation supplies a load-bearing uniqueness theorem, and the derivation chain is not shown to be equivalent to its inputs by construction. The conditions (regular variation, spectral positivity) are external to the claimed result and are standard in the literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard domain assumptions of probability theory concerning stable distributions and regular variation; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption Random variables lie in the basin of attraction of spectrally positive stable distributions under regular variation.
Invoked to apply the Tauberian theorem to the target families.

pith-pipeline@v0.9.0 · 5641 in / 1200 out tokens · 26367 ms · 2026-05-23T23:13:26.416697+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Kl¨ uppelberg and T

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Kulik and P

R. Kulik and P. Soulier. Heavy–Tailed Time Series . Springer, New York, 2020

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T. L. Lai. Uniform Tauberian theorems and their applications to r enewal theory and ﬁrst passage problems. Ann. Probab., 4:628–643, 1976. 28 GIAMPAOLO CRISTADORO AND GAIA POZZOLI

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