Geodesic
Application of number theory in mathematical statistics
Matematičeskie zametki, Tome 7 (1970) no. 4, pp. 383-388.

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Elementary number theory (divisibility theory) is used to prove that, in the case of repeated sampling, when the $k$-th generalized sample moment is independent of the sample mean then the sampling is normal if the volume of the sample is large and $k$ is a square-free integer which, if it is even, is not of the form $2(2s +1)$). 
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     author = {Yu. V. Linnik},
     title = {Application of number theory in mathematical statistics},
     journal = {Matemati\v{c}eskie zametki},
     pages = {383--388},
     publisher = {mathdoc},
     volume = {7},
     number = {4},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_7_4_a1/}
}
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Yu. V. Linnik. Application of number theory in mathematical statistics. Matematičeskie zametki, Tome 7 (1970) no. 4, pp. 383-388. http://geodesic.mathdoc.fr/item/MZM_1970_7_4_a1/