Path properties of dilatively stable processes and singularity of their distributions

TL;DR

This paper investigates the path regularity and distribution singularity of dilatively stable processes, revealing that processes with different parameters are mutually singular and establishing their local Hölder continuity properties.

Contribution

It provides new results on the Hölder continuity of dilatively stable processes and proves the singularity of their distributions when parameters differ.

Findings

Dilatively stable processes have almost surely constant local Hölder exponents.

Processes with different H parameters are mutually singular.

Results are specialized to limit processes and fractional Lévy processes.

Abstract

First, we present some results about the H\"older continuity of the sample paths of so called dilatively stable processes which are certain infinitely divisible processes having a more general scaling property than self-similarity. As a corollary, we obtain that the most important (H,delta)-dilatively stable limit processes (e.g., the LISOU and the LISCBI processes, see Igloi [4]) almost surely have a local H\"older exponent H. Next we prove that, under some slight regularity assumptions, any two dilatively stable processes with stationary increments are singular (in the sense that their distributions have disjoint supports) if their parameters H are different. We also study the more general case of not having stationary increments. Throughout the paper we specialize our results to some basic dilatively stable processes such as the above-mentioned limit processes and the fractional L\'evy process.