Random-Time Isotropic Fractional Stable Fields

Abstract

Generalizing both Substable Fractional Stable Motions (FSMs) and Indicator FSMs, we introduce α-stabilized subordination, a procedure which produces new FSMs (H-self-similar, stationary increment symmetric α-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 α-stabilized subordination. Finally we show that α-stabilized subordination of Linear FSMs produces null-conservative FSMs, a class of FSMs introduced by Samorodnitsky (Ann. Probab. 33(5):1782–1803, 2005).

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Notes

  1. 1.

    FSFs with H=max{1,1/α} are also possible, but other than α=1 they are unique [5, 24].

  2. 2.

    For stable processes, there are notions similar to covariances called covariations and codifferences, see [25, Chap. 2], and there is also a notion of spectral measure [14], but none of these is perfectly analogous to the beautiful characterization of Gaussian processes in terms of positive definition functions.

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Correspondence to Paul Jung.

Additional information

Research was started at Sogang University and supported in part by Sogang University Research Grant 201010073.

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Jung, P. Random-Time Isotropic Fractional Stable Fields. J Theor Probab 27, 618–633 (2014). https://doi.org/10.1007/s10959-012-0433-4

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Keywords

  • Fractional stable motion
  • Self-similar process
  • Stable field

Mathematics Subject Classification

  • 60G22
  • 60G52
  • 60G60