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Characterizations of the Normal Distribution by Suitable Transformations

S. Beer and E. Lukacs
Journal of Applied Probability
Vol. 10, No. 1 (Mar., 1973), pp. 100-108
DOI: 10.2307/3212498
Stable URL: http://www.jstor.org/stable/3212498
Page Count: 9
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Characterizations of the Normal Distribution by Suitable Transformations
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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.

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