Shot-noise processes with logarithmic response function and their scaling limits
Luisa Beghin, Lorenzo Cristofaro, Enrico Scalas
TL;DR
This work studies shot-noise processes whose impulse response depends on the logarithmic ratio of current time to event time, defined via $g(t/u)$ with slowly varying $g^*(y)$. It derives explicit finite-time covariances under several noise-variance structures and proves a scaling limit: the properly rescaled shot-noise converges to the Hadamard fractional Brownian motion $B^H_\alpha$ (with $\alpha\in(1,2)$), establishing a weak limit that combines Brownian marginals with long-range, non-stationary increments. The main contributions include closed-form finite-time covariances in key examples, a demonstration of ultra-long memory in the independent-noise setting, and a rigorous weak convergence to $B^H_\alpha$ using a Pan-type criterion in the Skorokhod $J_1$ topology. This Hadamard fBm acts as a middle ground between Brownian motion and classical fractional Brownian motion, with potential applications to aging systems and time-inhomogeneous diffusion models in physics and finance. The results provide a robust framework for modeling non-stationary, long-memory phenomena driven by log-ratio impulse responses.
Abstract
We consider shot-noise processes with an impulse response written in terms of the logarithm of the ratio between current and event time (instead of the usual absolute time difference). We study its finite-time properties as well as its weak convergence, under appropriate scaling and with general assumptions on the dependence of noises on event times. The limiting process coincides with the so-called Hadamard fractional Brownian motion (introduced in Beghin, Cristofaro, Polito (2026)), which represents a middle ground between standard Brownian motion and fractional Brownian motion. It shares the one-dimensional distribution with the former, while possessing the long-memory property (within a certain parameter range) of the latter, though with smaller intensity.