Large Time Asymptotic of Heavy Tailed Renewal Processes

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