Geodesic
On a~characterisation of a~class of probability distributions by those of some statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 479-487

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Let $\mathscr P$ be a class of probability distributions of random element $X$. The question is whether there exist a statistic $Y=f(X)$ possesing two following properties: 1$^\circ$. Its distribution $Q_P^Y$ under $P\in\mathscr P$ does not depend on $P$, $Q_P^Y=Q_\mathscr P^Y$. 2$^\circ$. If for some $P'$ $Q_{P'}^Y=Q_\mathscr P^Y$ then $P'\in\mathscr P$. The question is solved positively for some special families $\mathscr P$. 
@article{TVP_1965_10_3_a5,
     author = {Yu. V. Prokhorov},
     title = {On a~characterisation of a~class of probability distributions by those of some statistics},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {479--487},
     publisher = {mathdoc},
     volume = {10},
     number = {3},
     year = {1965},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1965_10_3_a5/}
}
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Yu. V. Prokhorov. On a~characterisation of a~class of probability distributions by those of some statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 479-487. http://geodesic.mathdoc.fr/item/TVP_1965_10_3_a5/