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Self-similar processes with independent increments
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  • Published: September 1991

Self-similar processes with independent increments

  • Ken-iti Sato1 

Probability Theory and Related Fields volume 89, pages 285–300 (1991)Cite this article

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Summary

A stochastic process {X t ∶t ≧0} onR d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X ct } and {aX t +b(t)} have common finite-dimensional distributions. If {X t } is widesense self-similar with independent increments, stochastically continuous, andX 0=const, then, for everyt, the distribution ofX t is of classL. Conversely, if μ is a distribution of classL, then, for everyH>0, there is a unique process {X t (H) } selfsimilar with exponentH with independent increments such thatX 1 has distribution μ. Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X t (H) } (called the process of classL with exponentH induced by μ) are compared with those of the Lévy process {Y t } such thatY 1 has distribution μ. Results are generalized to operator-self-similar processes and distributions of classOL. A process {X t } onR d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA c and a functionb c (t) such that {X ct } and {A c X t +b c (t)} have common finite-dimensional distributions. It is proved that, if {X t } is wide-sense operator-self-similar and stochastically continuous, then theA c can be chosen asA c =c Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason [4].

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Authors and Affiliations

  1. Department of Mathematics, College of General Education, Nagoya University, 464-01, Nagoya, Japan

    Ken-iti Sato

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  1. Ken-iti Sato
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Sato, Ki. Self-similar processes with independent increments. Probab. Th. Rel. Fields 89, 285–300 (1991). https://doi.org/10.1007/BF01198788

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  • Received: 13 March 1990

  • Revised: 18 October 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01198788

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Keywords

  • Stochastic Process
  • Probability Theory
  • Spectral Property
  • Mathematical Biology
  • Small Time
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