An elementary proof for dynamical scaling for certain fractional non-homogeneous Poisson processes

TL;DR

This paper provides a straightforward proof demonstrating the dynamical scaling property of certain fractional non-homogeneous Poisson processes, extending known results to a broader class of processes.

Contribution

It offers an elementary proof of dynamical scaling for fractional non-homogeneous Poisson processes, generalizing previous results and including the standard fractional Poisson process as a special case.

Findings

Dynamical scaling holds under mild conditions for these processes.

The proof encompasses the standard fractional Poisson process.

The result links fractional processes to self-similarity in physical systems.

Abstract

Dynamical scaling is an asymptotic property typical for the dynamics of first-order phase transitions in physical systems and related to self-similarity. Based on the integral-representation for the marginal probabilities of a fractional non-homogeneous Poisson process introduced by Leonenko et al. (2017) and generalising the standard fractional Poisson process, we prove the dynamical scaling under fairly mild conditions. Our result also includes the special case of the standard fractional Poisson process.