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Some dimension-free features of vector-valued martingales
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  • Published: June 1991

Some dimension-free features of vector-valued martingales

  • Olav Kallenberg1 &
  • Rafal Sztencel2 

Probability Theory and Related Fields volume 88, pages 215–247 (1991)Cite this article

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Summary

Given any local maringaleM inR d orl 2, there exists a local martingaleN inR 2, such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR 2 which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|2 and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.

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Author information

Authors and Affiliations

  1. Division of Mathematics, Auburn University, 120 Mathematics Annex, 36849-5307, Auburn, AL, USA

    Olav Kallenberg

  2. Department of Mathematics, Warsaw University, PL-00-901, Warsaw, PKiN, Poland

    Rafal Sztencel

Authors
  1. Olav Kallenberg
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  2. Rafal Sztencel
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Additional information

Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144

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Kallenberg, O., Sztencel, R. Some dimension-free features of vector-valued martingales. Probab. Th. Rel. Fields 88, 215–247 (1991). https://doi.org/10.1007/BF01212560

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  • Received: 30 July 1990

  • Revised: 27 October 1990

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01212560

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Keywords

  • Growth Rate
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Continuous Time
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