Random-Time Isotropic Fractional Stable Fields

Abstract

Generalizing both Substable Fractional Stable Motions (FSMs) and Indicator FSMs, we introduce alpha-stabilized subordination, a procedure which produces new FSMs (H-self-similar, stationary increment symmetric alpha-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 by Samorodnitsky (Ann. Probab. 33(5):1782-1803, 2005)