Dilatively semistable stochastic processes

TL;DR

This paper introduces dilative semistability, a new extension of semi-selfsimilarity for infinitely divisible stochastic processes, and characterizes these processes as limits of rescaled aggregations of independent processes.

Contribution

It defines dilative semistability, extends the concept of dilative stability, and provides characterizations and examples of such processes.

Findings

Dilative semistability generalizes semi-selfsimilarity.

Examples of dilatively semistable processes are provided.

Characterizations as limits of rescaled independent process aggregations.

Abstract

Dilative semistability extends the notion of semi-selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. It is shown that this scaling relation is a natural extension of dilative stability and some examples of dilatively semistable processes are given. We further characterize dilatively stable and dilatively semistable processes as limits for certain rescaled aggregations of independent processes.