Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions

TL;DR

This paper investigates the variance scaling in renewal-reward processes with heavy-tailed distributions, revealing anomalous behavior at specific power-law exponents and analyzing the effects in related physical models and many-body systems.

Contribution

It provides a detailed analysis of variance scaling in renewal-reward processes with heavy tails, identifying conditions for anomalous fluctuations and extending findings to physical models and many-body systems.

Findings

Anomalous variance scaling occurs at power-law exponent -3.

For exponents less than -3, fluctuations follow standard large deviation scaling.

Variance scaling becomes normal in many-body Knudsen gas despite boundary conditions with exponent -3.

Abstract

For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have reveals that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that the processes only fluctuate around the expectation with an error that is compatible with a standard large deviation scaling. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3.