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

TL;DR

This paper develops a perturbative expansion to better understand the distribution of sums of random variables with stretched-exponential tails, bridging the gap between typical fluctuations and rare large jumps, with applications to CTRWs.

Contribution

It introduces a systematic correction method extending the Big Jump Principle beyond asymptotic limits, applicable to continuous-time random walks with non-Gaussian statistics.

Findings

Explicit higher order corrections describe moderate deviations.

The approach bridges Gaussian fluctuations and big jump behavior.

Numerical simulations support analytical predictions.

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.