Geodesic
On S.\,U.~Bernstein's regularity type conditions in a~problem of empirical Bayes approach
Zapiski Nauchnykh Seminarov POMI, Studies in the statistical estimation theory. Part II, Tome 79 (1978), pp. 38-43

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It is proved that the aposteriorl distribution of a random success probability $X$ in the binomial scheme can be approximated by a suitable beta-distrubution if the number $n$ of trials tends to infinity and an apriori density function of $X$ belongs to $L^r[0,1]$ for some $r\geq1$. 
@article{ZNSL_1978_79_a3,
     author = {M. S. Nikulin},
     title = {On {S.\,U.~Bernstein's} regularity type conditions in a~problem of empirical {Bayes} approach},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {38--43},
     publisher = {mathdoc},
     volume = {79},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1978_79_a3/}
}
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M. S. Nikulin. On S.\,U.~Bernstein's regularity type conditions in a~problem of empirical Bayes approach. Zapiski Nauchnykh Seminarov POMI, Studies in the statistical estimation theory. Part II, Tome 79 (1978), pp. 38-43. http://geodesic.mathdoc.fr/item/ZNSL_1978_79_a3/