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

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.

22 pages, 6 figures

References in corpus (29)

• The large deviation approach to statistical mechanics
• Stochastic Resetting and Applications
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• The Nature of the Condensate in Mass Transport Models
• Large deviations for random walks under subexponentiality: the big-jump domain
• Packets of Diffusing Particles Exhibit Universal Exponential Tails
• The Single Big Jump Principle in Physical Modelling
• A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
• Large fluctuations and dynamic phase transition in a system of self-propelled particles
• Anomalous scaling of dynamical large deviations
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Rare events in generalized Lévy Walks and the Big Jump principle
• Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
• Dynamical large deviations of linear diffusions
• Anomalous scaling of dynamical large deviations of stationary Gaussian processes
• Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory
• Brownian non-Gaussian polymer diffusion and queing theory in the mean-field limit
• Polymers critical point originates Brownian non-Gaussian diffusion
• Rare events in stochastic processes with sub-exponential distributions and the Big Jump principle
• Fast rare events in exit times distributions of jump processes
• Development and regression of a large fluctuation
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Fractional Edgeworth Expansion: Corrections to the Gaussian-Lévy Central Limit Theorem
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• The case of the biased quenched trap model in two dimensions with diverging mean dwell times
• Rare Events in Extreme Value Statistics of Jump Processes with Power Tails
• Dynamical large deviations of the fractional Ornstein-Uhlenbeck process
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables

Cited by in corpus (1)

• Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain