Random-time isotropic fractional stable fields

TL;DR

This paper introduces alpha-stabilized subordination to generate new isotropic stable fields with stationary increments, extending existing FSMs and providing a Stable CLT, with applications to null-conservative FSMs.

Contribution

It develops alpha-stabilized subordination for FSMs, extends these to isotropic stable fields, and proves a Stable CLT, advancing the theory of stable processes and fields.

Findings

Alpha-stabilized subordination produces new FSMs from existing ones.

Extension to isotropic stable fields with stationary increments.

Stable CLT provides an intuitive understanding of the subordination process.

Abstract

Generalizing both Substable FSMs and Indicator FSMs, we introduce alpha-stabilized subordination, a procedure which produces new FSMs (H-sssi symmetric stable processes) from old ones. We extend these processes to isotropic stable fields which have stationary increments in the strong sense, i.e., processes which are invariant under Euclidean rigid motions of the multi-dimensional time parameter. We also prove a Stable Central Limit Theorem which provides an intuitive picture of alpha-stabilized subordination. Finally we show that alpha-stabilized subordination of Linear FSMs produces null-conservative FSMs, a class of FSMs introduced in Samorodnitsky (2005).