Dilatively stable stochastic processes and aggregate similarity

TL;DR

This paper explores dilatively stable processes, extending their definition and showing their equivalence to aggregate similarity in infinitely divisible processes, with examples and applications in aggregation models.

Contribution

It reformulates and extends the concept of dilative stability using characteristic functions and demonstrates the equivalence with aggregate similarity for infinitely divisible processes.

Findings

Dilatively stable processes generalize self-similar processes.

Dilatively stable generalized fractional Lévy processes are provided as examples.

Certain limit processes in aggregation models are dilatively stable.

Abstract

Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Igl\'oi (2008) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (2005). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional L\'evy processes are given and we point out that certain limit processes in aggregation models are dilatively stable.