Fractionally integrated inverse stable subordinators

TL;DR

This paper introduces the fractional integration of inverse stable subordinators (FIISS), demonstrating its role as a scaling limit in renewal shot noise processes with heavy-tailed distributions, and explores its regularity and asymptotic properties.

Contribution

It establishes FIISS as a new scaling limit for renewal shot noise processes with heavy tails and analyzes its regularity and asymptotic behavior.

Findings

FIISS is a scaling limit in the Skorokhod space for certain renewal processes.

Proves local Hölder continuity of FIISS.

Establishes a law of iterated logarithm for FIISS at small and large times.

Abstract

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of a power function and an inverse stable subordinator. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean `inter-shot' distribution and regularly varying response function. We prove local H\"{o}lder continuity of FIISS and a law of iterated logarithm for both small and large times.