Shot-noise processes with logarithmic response function and their scaling limits

TL;DR

This paper introduces a new class of shot-noise processes with a logarithmic response function, analyzes their finite-time behavior, and demonstrates their convergence to Hadamard fractional Brownian motion, linking finite-time models to complex long-memory processes.

Contribution

It presents a novel shot-noise process with a logarithmic response, establishing its scaling limit as Hadamard fractional Brownian motion, bridging finite-time models and long-memory stochastic processes.

Findings

The shot-noise process converges to Hadamard fractional Brownian motion under scaling.

The process exhibits long-memory properties within certain parameter ranges.

It provides a finite-time stochastic model for Hadamard fractional Brownian motion.

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 with the former the one-dimensional distribution (i.e. Gaussian with the same first two moments), while possessing the long-memory property (within a certain parameter range) of the latter, though with smaller intensity. Therefore, we identify a natural probabilistic scheme based on shot-noise processes whose scaling limit is the Hadamard fractional Brownian motion, thereby providing a concrete stochastic finite-time counterpart of this process.