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Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales
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  • Published: September 1991

Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales

  • Ditlev Monrad1 &
  • Walter Philipp1 

Probability Theory and Related Fields volume 88, pages 381–404 (1991)Cite this article

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Summary

In this paper we focus on sequences of random vectors which do not admit a strong approximation of their partial sums by sums of independent random vectors. In the first part we prove conditional versions of the Strassen-Dudley theorem. We apply these in the second part of the paper to obtain strong invariance principles for vector-valued martingales which, when properly normalized, converge in law to a mixture of Gaussian distributions.

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Author information

Authors and Affiliations

  1. Departments of Mathematics and Statistics, University of Illinois, 61801, Urbana, IL, USA

    Ditlev Monrad & Walter Philipp

Authors
  1. Ditlev Monrad
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  2. Walter Philipp
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Additional information

Research partially supported by the Air Force Office of Scientific Research Contract NO. F49260 85 C 0144

Part of this work was done while the second author was visiting the Department of Statistics and the Center for Stochastic Processes, University of North Carolina, Chapel Hill. He thanks the members of the Department of Statistics for their hospitality

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Cite this article

Monrad, D., Philipp, W. Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales. Probab. Th. Rel. Fields 88, 381–404 (1991). https://doi.org/10.1007/BF01418867

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  • Received: 11 June 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01418867

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Keywords

  • Gaussian Distribution
  • Hilbert Space
  • Stochastic Process
  • Probability Theory
  • Random Vector
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