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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Research supported by DFG – Forschergruppe FOR 399/2
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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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DOI: https://doi.org/10.1007/s00440-006-0503-6
Mathematics Subject Classification (1991)
- 62E10
- 62H05
Key words or phrases
- Gaussian random variable
- Independent random variables
- Entire characteristic functions
- Moments of random variables