Large-time asymptotic of heavy tailed renewal processes

TL;DR

This paper investigates the long-term behavior of renewal processes with heavy-tailed waiting times, revealing how singularities in large deviation functions lead to anomalous fluctuations and providing bounds for specific distributions.

Contribution

It introduces an expansion approach to analyze the emergence of singularities in large deviation functions for heavy-tailed renewal processes, with bounds for Pareto distributions.

Findings

Heavy tails cause extremely slow dynamics and singular large deviation functions.

The expansion approach provides an upper bound for the moment generating function.

Numerical simulations suggest similar behaviors in various heavy-tailed distributions.

Abstract

We study the large-time asymptotic of renewal-reward processes with a heavy-tailed waiting time distribution. It is known that the heavy tail of the distribution produces an extremely slow dynamics, resulting in a singular large deviation function. When the singularity takes place, the bottom of the large deviation function is flattened, manifesting anomalous fluctuations of the renewal-reward processes. In this article, we aim to study how these singularities emerge as the time increases. Using a classical result on the sum of random variables with regularly varying tail, we develop an expansion approach to prove an upper bound of the finite-time moment generating function for the Pareto waiting time distribution (power law) with an integer exponent. We perform numerical simulations using Pareto (with a real value exponent), inverse Rayleigh and log-normal waiting time distributions, and demonstrate similar results are anticipated in these waiting time distributions.