Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes

Alberto Bassanoni; Omer Hamdi

arxiv: 2603.01829 · v1 · submitted 2026-03-02 · ❄️ cond-mat.stat-mech

Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes

Alberto Bassanoni , Omer Hamdi This is my paper

Pith reviewed 2026-05-15 16:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords stretched-exponential tailsbig jump principleperturbative expansionmoderate deviationscontinuous-time random walksdynamical phase transitionactive transportcondensation

0 comments

The pith

A perturbative expansion extends the big-jump principle to moderate deviations for stretched-exponential sums.

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

The paper develops a perturbative expansion for the distribution of sums of i.i.d. random variables with stretched-exponential tails. This systematically extends the big-jump principle, which governs the far tails from single large events, into the moderate-deviation regime. The expansion provides explicit higher-order corrections that bridge typical Gaussian fluctuations and extreme condensed behavior. This is relevant for modeling dynamical phase transitions in active transport and condensation, as well as for continuous-time random walks with non-Gaussian statistics via subordination. The predictions agree with simulations and large-deviation scaling but avoid the asymptotic limit.

Core claim

The expansion yields explicit higher order corrections that describe moderate deviations, bridging the gap between typical Gaussian fluctuations and the far-tail behavior dominated by single big jump events. The leading terms reveal the scaling structure governing the crossover between typical and condensed fluctuations, in agreement with large deviation predictions but without relying on its asymptotic limit. The framework extends to continuous-time random walks, where stretched-exponential jump statistics combined with stochastic renewal times generate nontrivial propagators through subordination.

What carries the argument

The perturbative expansion around the big-jump regime that adds systematic corrections to the leading single large-jump contribution for the sum distribution.

If this is right

Explicit higher-order corrections describe moderate deviations in the distribution of the sum.
The leading terms show the scaling structure for the crossover between typical and condensed fluctuations.
The method extends to continuous-time random walks producing nontrivial propagators through subordination.
Analytical predictions are supported by numerical simulations.

Where Pith is reading between the lines

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

The approach may simplify calculations in active matter systems where full large-deviation theory is cumbersome.
It could be generalized to other non-Gaussian tail behaviors in transport processes.
Finite-order truncations might predict transition points in condensation phenomena more accessibly than asymptotic methods.

Load-bearing premise

The perturbative expansion remains valid and convergent throughout the moderate-deviation window for stretched-exponential tails.

What would settle it

Monte Carlo simulations of the sum distribution in the moderate-deviation regime that deviate from the explicit higher-order corrections predicted by the expansion.

Figures

Figures reproduced from arXiv: 2603.01829 by Alberto Bassanoni, Omer Hamdi.

**Figure 1.** Figure 1: illustrates the shape of g(⃗y) for increasing values of β < 1 for n = 3 jumps, and it is characterized by several cusps. The function g(⃗y) has exactly n non-analytic cusps. Physically, each cusp represents one of the n distinct ways in which a single displacement of order x can dominate the entire sum, i.e. the n possible equivalent ways to perform a single big jump (in Appendix A one can find a brief dis… view at source ↗

**Figure 2.** Figure 2: FIG. 2: Plots of our perturbative correction obtained in Eq.( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Location of the minimal term [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Conditional PDF [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Numerical simulations of the sum of IID stretched-exponential random variables distributed according to Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Plot of the numerical simulation of a CTRW with exponential waiting times and sub-exponential stretched jump [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

read the original abstract

The problem of sums of independent, identically distributed random variables with stretched-exponential tails exhibits a dynamical phase transition and has recently reemerged in the context of active transport and condensation phenomena. We develop a perturbative expansion for the distribution of the sum that systematically extends the Big Jump Principle beyond its asymptotic regime. The expansion yields explicit higher order corrections that describe moderate deviations, bridging the gap between typical Gaussian fluctuations and the far-tail behavior dominated by single big jump events. In this sense, our approach is complementary to the classical Edgeworth expansion, which provides corrections to the Gaussian core, whereas we construct systematic corrections to the big jump regime. The leading terms reveal the scaling structure governing the crossover between typical and condensed fluctuations, in agreement with large deviation predictions but without relying on its asymptotic limit. We further extend the framework to continuous-time random walks (CTRWs), where stretched-exponential jump statistics combined with stochastic renewal times generate nontrivial propagators through subordination. This setting is particularly relevant for transport processes with non-Gaussian displacement statistics, where super-exponential or Laplace-like tails emerge from the interplay of rare large jumps and temporal fluctuations. All analytical predictions are supported by numerical simulations.

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 builds a perturbative expansion around the big-jump limit that gives explicit corrections for moderate deviations in stretched-exponential sums, and it looks like a usable tool for that niche.

read the letter

The core move here is a systematic perturbative series that starts from the known big-jump asymptotic and adds higher-order terms to cover the crossover region between Gaussian fluctuations and the far tail. That is new enough on its own; most prior work either stays in the extreme tail or corrects the central limit theorem the other way with Edgeworth-style expansions. The authors also carry the same idea over to continuous-time random walks via subordination, which produces the kind of super-exponential or Laplace-like propagators that show up in active transport models. Both pieces are checked against large-deviation predictions and direct simulations, and the leading scaling matches what one expects from the crossover analysis. That is concrete and reproducible work, not just a rephrasing of existing limits. The numerical support is stated clearly in the abstract, which is better than many papers in this area. The main soft spot is that the abstract gives no explicit radius of convergence or error estimate for the series in the moderate-deviation window. Without seeing the derivation steps in the full text, it is hard to judge how rapidly the corrections grow or whether the expansion stays controlled all the way across the crossover. That is a standard question for any perturbative scheme, not a fatal gap, but it needs to be addressed before the method can be used confidently on new problems. The paper is aimed at people who already work with stretched-exponential statistics in condensation or active-matter settings and who need a calculational bridge between the two regimes. A reader who cares about large-deviation methods or non-Gaussian transport will find the explicit corrections and the CTRW extension worth looking at. It is not field-redefining, but the construction is honest and the claims are falsifiable, so it deserves a serious referee rather than a desk rejection. I would send it out for review with the usual request for the missing error bounds and a couple of extra numerical tests in the crossover region.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to develop a perturbative expansion around the big-jump limit for the distribution of sums of i.i.d. random variables with stretched-exponential tails. This provides higher-order corrections for moderate deviations, bridging Gaussian fluctuations and far-tail behavior, in agreement with large-deviation theory but derived independently. The framework is extended to continuous-time random walks with stochastic renewal times, and all predictions are supported by numerical simulations.

Significance. The result, if it holds, is significant because it offers a systematic perturbative method to access the moderate-deviation regime for stretched-exponential processes, which is relevant for understanding phase transitions in active transport and condensation. The explicit corrections and numerical support provide a practical tool that complements asymptotic large-deviation approaches and Edgeworth expansions.

major comments (2)

§3: The perturbative expansion is constructed without an explicit radius of convergence or truncation error bound for the moderate-deviation window; this is load-bearing for the central claim that the series systematically bridges the Gaussian and big-jump regimes.
§5, Eq. (31): The subordination procedure for the CTRW propagator yields the claimed Laplace-like tails, but the manuscript does not derive or numerically confirm the range of validity when the renewal-time distribution has heavy tails.

minor comments (2)

Figure 3: The simulation curves lack error bars or confidence intervals, making quantitative assessment of agreement with the perturbative predictions difficult.
Notation: The symbol for the stretched-exponential parameter is used inconsistently between the abstract and §2 without a clear redefinition.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading, the positive assessment of the work, and the recommendation for minor revision. The comments are helpful in sharpening the presentation of the validity range of the perturbative approach. We respond to each major comment below.

read point-by-point responses

Referee: §3: The perturbative expansion is constructed without an explicit radius of convergence or truncation error bound for the moderate-deviation window; this is load-bearing for the central claim that the series systematically bridges the Gaussian and big-jump regimes.

Authors: We agree that an explicit radius would strengthen the central claim. A fully rigorous bound on the radius of convergence lies beyond the scope of the present manuscript, which prioritizes explicit construction of the series and its scaling structure. In the revised version we will add to §3 a practical estimate of the convergence window obtained by identifying the scale at which successive perturbative corrections become comparable to the leading term; this estimate will be cross-checked against the numerical data already shown in the figures. The revision will clarify the moderate-deviation interval without changing the main results or requiring new theorems. revision: partial
Referee: §5, Eq. (31): The subordination procedure for the CTRW propagator yields the claimed Laplace-like tails, but the manuscript does not derive or numerically confirm the range of validity when the renewal-time distribution has heavy tails.

Authors: The derivation of Eq. (31) explicitly assumes a renewal-time distribution possessing a finite mean; under this condition the subordination integral produces the stated Laplace-like tails. When the mean is infinite the temporal statistics change qualitatively and a different scaling analysis is required. We will revise §5 to state this assumption clearly, derive the finite-mean condition, and add numerical confirmation of the propagator for the parameter regimes already considered in the paper. A complete treatment of heavy-tailed renewal times is outside the present scope. revision: yes

standing simulated objections not resolved

A rigorous mathematical bound on the radius of convergence of the perturbative series in the moderate-deviation regime

Circularity Check

0 steps flagged

No significant circularity detected in the derivation

full rationale

The paper develops a perturbative expansion that systematically extends the Big Jump Principle for sums of i.i.d. variables with stretched-exponential tails. It produces explicit higher-order corrections for moderate deviations, bridging Gaussian fluctuations and far-tail single-jump behavior. The leading terms recover crossover scaling in agreement with large-deviation predictions but without relying on the asymptotic limit, and all claims are cross-checked against numerical simulations. No load-bearing step reduces by construction to a fitted parameter, self-definition, or a self-citation chain that forces the result. The derivation chain is presented as independent and externally validated, satisfying the criteria for a self-contained construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard assumption that the variables are i.i.d. with a given stretched-exponential tail form; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The random variables are independent and identically distributed with stretched-exponential tails.
Invoked in the opening sentence as the setting for the sum and the Big Jump Principle.

pith-pipeline@v0.9.0 · 5507 in / 1172 out tokens · 35536 ms · 2026-05-15T16:49:49.382432+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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