Exponents of operator self-similar random fields

TL;DR

This paper investigates the properties and relationships of exponents in operator self-similar random fields, providing a characterization of the possible exponents E and H that define their self-similarity.

Contribution

It characterizes the sets of possible range and domain exponents for operator self-similar random fields, clarifying the non-uniqueness due to symmetry.

Findings

Characterization of the set of range exponents H for a fixed domain exponent E.

Characterization of the set of domain exponents E for a fixed range exponent H.

Insight into the symmetry and non-uniqueness of exponents in operator self-similarity.

Abstract

If X(c^E t) and c^H X(t) have the same finite-dimensional distributions for some linear operators E and H, we say that the random vector field X(t) is operator self-similar. The exponents E and H are not unique in general, due to symmetry. This paper characterizes the possible set of range exponents H for a given domain exponent, and conversely, the set of domain exponents E for a given range exponent.