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Independence of linear forms with random coefficients
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  • Published: 24 April 2006

Independence of linear forms with random coefficients

  • G. P. Chistyakov1 &
  • F. Götze2 

Probability Theory and Related Fields volume 137, pages 1–24 (2007)Cite this article

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Abstract

We extend the classical Darmois–Skitovich theorem to the case where the linear forms L r 1 = U 1 X 1+ · · · +U n X n and L r 2 = U n +1 X 1+· · · +U 2 n X n have random coefficients U 1, . . . ,U 2 n . Under minimal restrictions on the random coefficients we completely describe the distributions of the independent random variables X 1, . . . ,X n and U 1, . . . ,U 2 n such that the linear forms L r 1 and L r 2 are independent.

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Author information

Authors and Affiliations

  1. Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave, 61103, Kharkov, Ukraine

    G. P. Chistyakov

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    F. Götze

Authors
  1. G. P. Chistyakov
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  2. F. Götze
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Corresponding author

Correspondence to F. Götze.

Additional information

Research supported by DFG – Forschergruppe FOR 399/2

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Cite this article

Chistyakov, G., Götze, F. Independence of linear forms with random coefficients. Probab. Theory Relat. Fields 137, 1–24 (2007). https://doi.org/10.1007/s00440-006-0503-6

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  • Received: 10 March 2004

  • Revised: 02 February 2005

  • Published: 24 April 2006

  • Issue Date: January 2007

  • DOI: https://doi.org/10.1007/s00440-006-0503-6

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Mathematics Subject Classification (1991)

  • 62E10
  • 62H05

Key words or phrases

  • Gaussian random variable
  • Independent random variables
  • Entire characteristic functions
  • Moments of random variables
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