Characterizations of the normal distribution by suitable transformations
- 1 March 1973
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 10 (01) , 100-108
- https://doi.org/10.1017/s002190020004211x
Abstract
Yu. V. Linnik showed that certain transformations, given by Formulae (1.1), (1.6) and (1.7) transform a normal sample into itself. The transformations (1.1) and (1.7) apply to samples of size 2 while (1.6) admits an arbitrary sample size. It is also assumed that the population mean is zero. In the present paper the converse theorems are proven so that characterizations of the normal distribution are obtained. The problem leads to the functional equations (2.3) and (2.13) whose solution yields the desired results.Keywords
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