A probability distribution on an $n$-dimensional Euclidean space can be specified by giving either the joint distribution of the usual rectangular coordinates or by the joint distribution of the polar coordinates. The object of this note is to obtain a characterization of normality in terms of independence of these coordinates under suitable regularity conditions. Further, if we slightly relax these conditions we get a new class of distributions which includes the normal distribution.
"A Characterization of Normality." Ann. Math. Statist. 38 (6) 1924 - 1927, December, 1967. https://doi.org/10.1214/aoms/1177698630