According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space.
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.
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Myronyuk, M.V. On the Skitovich-Darmois theorem and Heyde theorem in a Banach space. Ukr Math J 60, 1437–1447 (2008). https://doi.org/10.1007/s11253-009-0142-z
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DOI: https://doi.org/10.1007/s11253-009-0142-z