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Infinite Divisibility for Stochastic Processes and Time Change

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General results concerning infinite divisibility, selfdecomposability, and the class L m property as properties of stochastic processes are presented. A new concept called temporal selfdecomposability of stochastic processes is introduced. Lévy processes, additive processes, selfsimilar processes, and stationary processes of Ornstein–Uhlenbeck type are studied in relation to these concepts. Further, time change of stochastic processes is studied, where chronometers (stochastic processes that serve to change time) and base processes (processes to be time-changed) are independent but do not, in general, have independent increments. Conditions for inheritance of infinite divisibility and selfdecomposability under time change are given.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000, Aarhus C, Denmark

    Ole E. Barndorff-Nielsen

  2. Department of Mathematics, Keio University, Hiyoshi, Yokohama, 223-8522, Japan

    Makoto Maejima

  3. Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074, Japan

    Ken-iti Sato

Authors
  1. Ole E. Barndorff-Nielsen
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  2. Makoto Maejima
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  3. Ken-iti Sato
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Correspondence to Ken-iti Sato.

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Barndorff-Nielsen, O.E., Maejima, M. & Sato, Ki. Infinite Divisibility for Stochastic Processes and Time Change. J Theor Probab 19, 411–446 (2006). https://doi.org/10.1007/s10959-006-0020-7

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  • DOI: https://doi.org/10.1007/s10959-006-0020-7

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