Geodesic
Necessary and Sufficient Statistics for the Family of Shifts of Probability Distributions on Continious Bicompact Groups
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 307-321 
V. M. Maksimov. Necessary and Sufficient Statistics for the Family of Shifts of Probability Distributions on Continious Bicompact Groups. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 307-321. http://geodesic.mathdoc.fr/item/TVP_1967_12_2_a5/
@article{TVP_1967_12_2_a5,
     author = {V. M. Maksimov},
     title = {Necessary and {Sufficient} {Statistics} for the {Family} of {Shifts} of {Probability} {Distributions} on {Continious} {Bicompact} {Groups}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {307--321},
     year = {1967},
     volume = {12},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_2_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that if the family of probability distributions $p_y(x)=p(xy)$ on a bicompact group $G$, where $x\in G$ and $y\in G$, has nontrivial sufficient statistics for the parameter $у$ then the density $p(x)$ may be written in the form $p(x)=\exp\Bigl(\lambda+\sum_{k=1}^sc_k\varphi_k(x)\Bigr)$ where $1,\varphi_1(x),\dots,\varphi_s(x)$ is a basis of the set of entries of the matrix $\{g_{ij}(x)\}$ of a certain real finite-dimensional representation of group $G$. The case when $p(x)$ may be equal to zero is also considered (here we deal mainly with the case when $G$ is the circle group).